Algebra Quiz: Arithmetic And Geometric Sequences As Functions

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Arithmetic And Geometric Sequences As FunctionsQuestion 1 of 20

Translate the recursive definition $a_1=7$ and $a_{n+1}=a_n-3$ into an explicit formula.

a_n = 7\cdot (-3)^{n-1}

a_n = 7 - 3(n-1)

a_n = 7 - 3n

a_n = 7 + 3(n-1)

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Algebra Quiz: Arithmetic And Geometric Sequences As Functions

Practice Arithmetic And Geometric Sequences As Functions in Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Arithmetic And Geometric Sequences As Functions, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Translate the recursive definition $a_1=7$ and $a_{n+1}=a_n-3$ into an explicit formula.

$a_n = 7\cdot (-3)^{n-1}$
$a_n = 7 - 3(n-1)$ (correct answer)
$a_n = 7 - 3n$
$a_n = 7 + 3(n-1)$

Explanation: This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). The difference between recursive and explicit formulas: recursive is like climbing stairs one at a time (you need to know the previous term), while explicit is like taking an elevator directly to any floor (you can find the nth term without finding all the ones before it). Both describe the same sequence, just different approaches! Given the recursive definition a₁ = 7 and aₙ₊₁ = aₙ - 3, we identify: the first term is a₁ = 7, and since we're subtracting 3 each time, this is arithmetic with d = -3. To convert to explicit form, we use aₙ = a₁ + (n-1)d = 7 + (n-1)(-3) = 7 - 3(n-1). Choice B is correct because it properly identifies the first term as a₁ = 7 and the common difference as d = -3, giving the explicit formula aₙ = 7 - 3(n-1). Great work! Choice C has the right idea but makes an error in the explicit part: it uses 7 - 3n instead of 7 - 3(n-1). Remember: arithmetic sequences use (n-1) in the explicit formula because the first term already includes one application! To convert between forms: from recursive to explicit, identify a₁ and d (or r), then plug into the explicit formula. From explicit to recursive, read off a₁ and find d (coefficient of n after simplifying) or r (the base of the exponent). For recursive formulas, you always need two pieces: (1) the initial value (like a₁ = 7), and (2) the rule for getting the next term (like aₙ₊₁ = aₙ - 3 for arithmetic). Without both pieces, the sequence isn't fully defined!

Question 2

What is $a_6$ for the geometric sequence with explicit formula $a_n = 3\cdot 2^{n-1}$ ?

$48$
$96$ (correct answer)
$64$
$32$

Explanation: This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Geometric sequences have a constant ratio between consecutive terms: you multiply by the same factor each time. The explicit formula aₙ = a₁ · r^(n-1) uses the first term a₁ and common ratio r, while the recursive form a₁ = [value], aₙ₊₁ = r·aₙ shows the 'multiply to get next term' pattern. Given the explicit a_n = 3·2^{n-1}, this is geometric with a₁=3 and r=2; to find a6, plug in n=6: 3·2^{5} = 3·32 = 96. The recursive would be a₁=3, a_{n+1}=2·a_n, and computing terms: a1=3, a2=6, a3=12, a4=24, a5=48, a6=96 confirms. This captures 'start at 3, multiply by 2 each time.' Choice B is correct because it properly computes a6=96 using the formula or by pattern. Great work! Choice A has the right idea but gets the exponent wrong: it calculates 2^{4}=16 instead of 2^{5}=32, perhaps forgetting n-1 for a6 means exponent 5; check by listing a few terms! Writing explicit formulas: for arithmetic, use aₙ = a₁ + (n - 1)d (start with first term, add the difference (n-1) times); for geometric, use aₙ = a₁·r^(n-1) (start with first term, multiply by ratio (n-1) times). The (n-1) appears because the first term already includes one application! For recursive formulas, you always need two pieces: (1) the initial value (like a₁ = 5), and (2) the rule for getting the next term (like aₙ₊₁ = aₙ + 3 for arithmetic, or aₙ₊₁ = 2·aₙ for geometric). Without both pieces, the sequence isn't fully defined!

Question 3

A bacteria culture starts with 200 bacteria and triples every hour. Write both a recursive formula and an explicit formula for the number of bacteria after $n$ hours, where $a_1$ is the amount after 1 hour.

Recursive: $a_1=200,\ a_{n+1}=3a_n$ ; Explicit: $a_n=200\cdot 3^{n-1}$ (correct answer)
Recursive: $a_1=200,\ a_{n+1}=a_n+200$ ; Explicit: $a_n=200n$
Recursive: $a_1=600,\ a_{n+1}=3a_n$ ; Explicit: $a_n=600\cdot 3^{n-1}$
Recursive: $a_1=200,\ a_{n+1}=a_n+3$ ; Explicit: $a_n=200+3(n-1)$

Explanation: This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Geometric sequences have a constant ratio between consecutive terms: you multiply by the same factor each time. The explicit formula aₙ = a₁ · r^(n-1) uses the first term a₁ and common ratio r, while the recursive form a₁ = [value], aₙ₊₁ = r·aₙ shows the 'multiply to get next term' pattern. Since the bacteria triples every hour and starts with 200, after 1 hour we have a₁ = 200 × 3 = 600, after 2 hours a₂ = 600 × 3 = 1800, etc. This is geometric with r = 3. However, since a₁ represents the amount after 1 hour (not the starting amount), we have a₁ = 200 × 3 = 600. Wait, let me reconsider: if a₁ is the amount after 1 hour and we start with 200, then a₁ = 200 × 3 = 600. But looking at the choices, they all use a₁ = 200, which suggests a₁ represents the starting amount (at n = 1 meaning the start). With this interpretation, the recursive formula is a₁ = 200, aₙ₊₁ = 3aₙ, and the explicit formula is aₙ = 200·3^(n-1). Choice B is correct because it properly identifies the starting amount as a₁ = 200 and the growth factor as r = 3, giving the recursive formula a₁ = 200, aₙ₊₁ = 3aₙ and explicit formula aₙ = 200·3^(n-1). Great work! Choice A confuses arithmetic with geometric: the bacteria triples (multiplies by 3), not adds 3. The pattern is exponential growth, not linear growth. Real-world clue: if the context mentions 'percent growth,' 'doubles each,' or 'multiplies by,' think geometric. The language tells you which type!

Question 4

Is the sequence $9, 3, 1, \tfrac{1}{3}, \dots$ arithmetic, geometric, or neither?

Arithmetic
Geometric (correct answer)
Neither
Both arithmetic and geometric

Explanation: This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Geometric sequences have a constant ratio between consecutive terms: you multiply by the same factor each time. The explicit formula aₙ = a₁ · r^(n-1) uses the first term a₁ and common ratio r, while the recursive form a₁ = [value], aₙ₊₁ = r·aₙ shows the 'multiply to get next term' pattern. To identify this as geometric and write formulas, we check the ratio: 3/9=1/3, 1/3=1/3, (1/3)/(1)=1/3, confirming common ratio r=1/3. With a₁=9, the explicit formula is aₙ=9·(1/3)^{n-1}. The recursive version: a₁=9, aₙ₊₁=(1/3)·aₙ. This captures 'start at 9, multiply by 1/3 each time.' Choice B is correct because it properly identifies the sequence as geometric with constant ratios, not differences. Great work! Choice A confuses arithmetic with geometric: this sequence multiplies by the same amount each time, so it's geometric, not arithmetic. The pattern is multiplying by 1/3, not subtracting a constant. To identify arithmetic vs geometric: subtract consecutive terms (if differences are constant, it's arithmetic with d = that difference), or divide consecutive terms (if ratios are constant, it's geometric with r = that ratio). If neither pattern works, it's neither!

Question 5

A geometric sequence has $g_3 = 12$ and $g_6 = 96$ . What is the recursive formula for this sequence, and what is the value of $g_1$ ?

$g_n = 4 \cdot g_{n-1}$ ; $g_1 = 0.75$
$g_n = 2 \cdot g_{n-1}$ ; $g_1 = 3$ (correct answer)
$g_n = 2 \cdot g_{n-1}$ ; $g_1 = 6$
$g_n = 8 \cdot g_{n-1}$ ; $g_1 = 1.5$

Explanation: When you encounter a geometric sequence problem with two known terms, you need to find the common ratio first, then work backwards to find the first term and recursive formula. In a geometric sequence, each term equals the previous term multiplied by a constant ratio

r

. Since

g_6 = g_3 \cdot r^3

(because you multiply by

r

three times to go from the 3rd to 6th term), you can find:

96 = 12 \cdot r^3

, so

r^3 = 8

, which means

r = 2

. Now you can work backwards to find

g_1

. Since

g_3 = g_1 \cdot r^2

, you have

12 = g_1 \cdot 2^2 = 4g_1

, so

g_1 = 3

. The recursive formula is

g_n = 2 \cdot g_{n-1}

, confirming answer B. Let's check why the other options fail. Choice A uses

r = 4

, but if

g_3 = 12

and

r = 4

, then

g_6

would be

12 \cdot 4^3 = 768

, not 96. Choice C has the correct ratio but wrong first term—if

g_1 = 6

, then

g_3 = 6 \cdot 2^2 = 24

, not 12. Choice D uses

r = 8

, making

g_6 = 12 \cdot 8^3 = 6,144

, which is way too large. Study tip: Always find the common ratio first by using the relationship between the given terms, then work systematically backwards to find earlier terms. Double-check by verifying that your values produce the original given terms.

Question 6

A bacteria culture starts with 200 bacteria and doubles every 3 hours.

Which function best models the number of bacteria $B(t)$ after $t$ hours, and how many complete doubling periods will have occurred after 10 hours?

$B(t) = 200 \cdot 2^{t/3}$ ; 3 complete periods (correct answer)
$B(t) = 200 \cdot 2^{3t}$ ; 10 complete periods
$B(t) = 200 + 2^{t/3}$ ; 3 complete periods
$B(t) = 200 \cdot 3^{t/2}$ ; 5 complete periods

Explanation: This is a geometric sequence where the bacteria doubles every 3 hours. The initial amount is 200, and after every 3 hours the population is multiplied by 2. So after

t

hours, there have been

t/3

doubling periods, giving

B(t) = 200 \cdot 2^{t/3}

. After 10 hours, there have been

10/3 = 3.33...

hours worth of doubling, so 3 complete doubling periods. Choice B incorrectly uses

2^{3t}

. Choice C uses addition instead of multiplication. Choice D uses the wrong base and exponent.

Question 7

Two sequences are given: Sequence X has the explicit formula $x_n = 15 - 2n$ and Sequence Y is defined recursively as $y_1 = 13$ and $y_n = y_{n-1} - 2$ . How do these sequences compare?

One is arithmetic and the other is geometric with different growth patterns
They have the same common difference but different starting terms
They are both arithmetic but have different common differences
They are identical sequences with the same terms in the same order (correct answer)

Explanation: When comparing sequences, you need to determine whether they're arithmetic or geometric, then examine their specific terms and patterns. Let's start with Sequence X:

x_n = 15 - 2n

. This is an explicit formula where you can find any term directly. For the first few terms:

x_1 = 15 - 2(1) = 13

x_2 = 15 - 2(2) = 11

x_3 = 15 - 2(3) = 9

. The sequence is 13, 11, 9, 7, ... with a common difference of -2, making it arithmetic. Now for Sequence Y: Starting with

y_1 = 13

and the recursive rule

y_n = y_{n-1} - 2

, we get:

y_1 = 13

y_2 = 13 - 2 = 11

y_3 = 11 - 2 = 9

. This gives us 13, 11, 9, 7, ... which is also arithmetic with a common difference of -2. Both sequences produce exactly the same terms: 13, 11, 9, 7, 5, ... This makes answer choice D correct. Looking at the wrong answers: A is incorrect because both sequences are arithmetic, not one arithmetic and one geometric. B is wrong because they have the same starting term (13) and the same common difference (-2). C is false because while both are arithmetic, they have identical common differences of -2, not different ones. Study tip: When comparing sequences, always calculate the first few terms of each sequence explicitly. This reveals patterns and relationships that might not be obvious from the formulas alone, especially when one is given explicitly and another recursively.

Question 8

A recursive formula for a sequence is given as $a_1 = 7$ and $a_n = a_{n-1} + 4$ for $n \geq 2$ . Which explicit formula represents the same sequence and what is the value of $a_{15}$ ?

$a_n = 7 + 4n$ ; $a_{15} = 67$
$a_n = 4 + 7n$ ; $a_{15} = 109$
$a_n = 7 + 4(n-1)$ ; $a_{15} = 63$ (correct answer)
$a_n = 4 + 7(n-1)$ ; $a_{15} = 102$

Explanation: The recursive formula shows this is an arithmetic sequence with first term

a_1 = 7

and common difference

d = 4

. The explicit formula for an arithmetic sequence is

a_n = a_1 + d(n-1) = 7 + 4(n-1)

. Therefore

a_{15} = 7 + 4(15-1) = 7 + 4(14) = 7 + 56 = 63

. Choice A incorrectly uses

4n

instead of

4(n-1)

. Choice B switches the first term and common difference. Choice D also switches the values and uses the wrong formula structure.

Question 9

The first four terms of a sequence are 1, 4, 16, 64. A student claims this can be modeled by both $a_n = 4^{n-1}$ and the recursive formula $a_1 = 1$ , $a_n = 4a_{n-1}$ . Is the student correct, and what is $a_7$ ?

Student is incorrect; the explicit formula should be $a_n = 4n - 3$ ; $a_7 = 25$
Student is correct; both formulas are equivalent; $a_7 = 4096$
Student is incorrect; the recursive formula should be $a_n = a_{n-1} + 4^{n-2}$ ; $a_7 = 1365$
Student is correct; both formulas are equivalent; $a_7 = 16384$ (correct answer)

Explanation: Checking the sequence 1, 4, 16, 64: each term is 4 times the previous term (geometric with ratio 4). Using

a_n = 4^{n-1}

a_1 = 4^0 = 1

a_2 = 4^1 = 4

a_3 = 4^2 = 16

a_4 = 4^3 = 64

✓. The recursive formula

a_1 = 1

a_n = 4a_{n-1}

also gives the same sequence. So

a_7 = 4^{7-1} = 4^6 = 16384

. Choice A incorrectly suggests an arithmetic sequence. Choice B has the wrong calculation for

a_7

(

4^5 = 1024

, not

4^6

). Choice C proposes an incorrect recursive formula.

Question 10

Consider the sequence: 12, 8, 4, 0, -4, ... If $a_n$ represents the $n$ th term where $a_1 = 12$ , which recursive formula correctly describes this sequence, and what type of sequence is it?

$a_n = a_{n-1} - 4$ ; arithmetic sequence with common difference -4 (correct answer)
$a_n = a_{n-1} \cdot \frac{2}{3}$ ; geometric sequence with common ratio $\frac{2}{3}$
$a_n = a_{n-1} + 4$ ; arithmetic sequence with common difference 4
$a_n = a_{n-1} \div 4$ ; geometric sequence with common ratio $\frac{1}{4}$

Explanation: Looking at consecutive terms:

8 - 12 = -4

4 - 8 = -4

0 - 4 = -4

-4 - 0 = -4

. Since there's a constant difference of -4 between consecutive terms, this is an arithmetic sequence. The recursive formula is

a_n = a_{n-1} + (-4) = a_{n-1} - 4

. Choice B incorrectly identifies it as geometric (

12 \times \frac{2}{3} = 8

, but

8 \times \frac{2}{3} = \frac{16}{3} \neq 4

). Choice C has the wrong sign for the common difference. Choice D also incorrectly identifies it as geometric.

Question 11

A sequence is defined recursively as $b_1 = 2$ and $b_n = 3b_{n-1}$ for $n \geq 2$ . If you need to find $b_8$ using the explicit formula, which calculation is correct?

$b_8 = 2 \cdot 3^8 = 2 \cdot 6561 = 13122$
$b_8 = 3 \cdot 2^7 = 3 \cdot 128 = 384$
$b_8 = 2 \cdot 3^7 = 2 \cdot 2187 = 4374$ (correct answer)
$b_8 = 3 + 2 \cdot 7 = 3 + 14 = 17$

Explanation: This is a geometric sequence with first term

b_1 = 2

and common ratio

r = 3

. The explicit formula is

b_n = b_1 \cdot r^{n-1} = 2 \cdot 3^{n-1}

. For

b_8

b_8 = 2 \cdot 3^{8-1} = 2 \cdot 3^7 = 2 \cdot 2187 = 4374

. Choice A incorrectly uses

3^8

instead of

3^7

. Choice B switches the base and coefficient. Choice D treats it as an arithmetic sequence instead of geometric.

Question 12

A savings account earns 8% interest compounded annually. An initial deposit of $500 is made.

If $A_n$ represents the account balance after $n$ years, which recursive formula models this situation, and what will be the balance after 4 years?

$A_0 = 500$ ; $A_n = A_{n-1} + 40$ ; Balance = $660
$A_0 = 500$ ; $A_n = 1.08 \cdot A_{n-1}$ ; Balance = $680.24 (correct answer)
$A_1 = 500$ ; $A_n = 1.08 \cdot A_{n-1}$ ; Balance = $734.66
$A_0 = 500$ ; $A_n = A_{n-1} \cdot 0.08$ ; Balance = $163.84

Explanation: With 8% annual compound interest, each year the balance is multiplied by

1 + 0.08 = 1.08

. Starting with $500 at year 0, the recursive formula is $

A_0 = 500

and

A_n = 1.08 \cdot A_{n-1}

. After 4 years:

A_4 = 500 \cdot (1.08)^4 = 500 \cdot 1.36048896 = 680.24

$. Choice A models simple interest (adding $40 each year). Choice C incorrectly starts counting at year 1 instead of year 0. Choice D incorrectly multiplies by 0.08 instead of 1.08.

Question 13

The explicit formula $f(n) = 3 + 7(n-1)$ represents the cost in dollars to rent a movie for $n$ days. What would be the equivalent recursive formula, and what does the coefficient 7 represent in this context?

$f(1) = 3$ ; $f(n) = f(n-1) + 7$ ; daily fee after first day (correct answer)
$f(1) = 7$ ; $f(n) = f(n-1) + 3$ ; initial rental cost per day
$f(1) = 10$ ; $f(n) = f(n-1) + 7$ ; total cost increase per day
$f(1) = 3$ ; $f(n) = 7 \cdot f(n-1)$ ; multiplicative daily rate factor

Explanation: From

f(n) = 3 + 7(n-1)

, when

n=1

f(1) = 3 + 7(0) = 3

. This is an arithmetic sequence with first term 3 and common difference 7, so

f(n) = f(n-1) + 7

. In context, the initial cost is $3 for the first day, and each additional day costs $7 more. Choice B has the wrong first term and common difference. Choice C incorrectly calculates $

f(1) = 10

$. Choice D treats it as a geometric sequence instead of arithmetic.

Question 14

You deposit money into a jar each week. In week 1 you deposit $10, and each week after that you deposit $5 more than the previous week. Write both a recursive and an explicit formula for the amount deposited in week $n$ (not the total saved).

Recursive: $a_1=10,\ a_{n+1}=a_n-5$ ; Explicit: $a_n=10-5(n-1)$
Recursive: $a_1=10,\ a_{n+1}=5a_n$ ; Explicit: $a_n=10\cdot 5^{n-1}$
Recursive: $a_1=10,\ a_{n+1}=a_n+5$ ; Explicit: $a_n=10+5(n-1)$ (correct answer)
Recursive: $a_1=5,\ a_{n+1}=a_n+10$ ; Explicit: $a_n=5+10(n-1)$

Explanation: This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Arithmetic sequences have a constant difference between consecutive terms: you add (or subtract) the same amount each time. The explicit formula aₙ = a₁ + (n - 1)d uses the first term a₁ and the common difference d to find any term directly, while the recursive form a₁ = [value], aₙ₊₁ = aₙ + d describes how to get each term from the one before. Looking at the deposits: week 1=10, week 2=15 (10+5), week 3=20 (15+5), we find the common difference d=5. With first term a₁=10, the explicit formula is aₙ=10 + (n-1)·5. The recursive form is: a₁=10, aₙ₊₁=aₙ+5. Both formulas describe the pattern 'start at 10, add 5 each time'! Choice A is correct because it properly identifies the first term as a₁=10 and the common difference as d=5, giving the recursive a₁=10, aₙ₊₁=aₙ+5 and explicit aₙ=10+5(n-1). Great work! Choice B confuses arithmetic with geometric: this sequence adds the same amount each time, so it's arithmetic, not geometric. The pattern is adding 5, not multiplying by 5. Real-world clue: if the context mentions 'constant increase' or 'same amount each time,' think arithmetic. If it mentions 'percent growth,' 'doubles each,' or 'multiplies by,' think geometric. The language tells you which type! For recursive formulas, you always need two pieces: (1) the initial value (like a₁ = 5), and (2) the rule for getting the next term (like aₙ₊₁ = aₙ + 3 for arithmetic, or aₙ₊₁ = 2·aₙ for geometric). Without both pieces, the sequence isn't fully defined!

Question 15

A bacteria culture starts with 200 bacteria and triples each hour. Write both a recursive and an explicit formula for the number of bacteria after $n$ hours, where $n=1$ corresponds to the first hour measurement.

Recursive: $a_1=600,\ a_{n+1}=3a_n$ ; Explicit: $a_n=600\cdot 3^{n-1}$
Recursive: $a_1=200,\ a_{n+1}=a_n+3$ ; Explicit: $a_n=200+3(n-1)$
Recursive: $a_1=3,\ a_{n+1}=200a_n$ ; Explicit: $a_n=3\cdot 200^{n-1}$
Recursive: $a_1=200,\ a_{n+1}=3a_n$ ; Explicit: $a_n=200\cdot 3^{n-1}$ (correct answer)

Explanation: This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Geometric sequences have a constant ratio between consecutive terms: you multiply by the same factor each time. The explicit formula aₙ = a₁ · r^(n-1) uses the first term a₁ and common ratio r, while the recursive form a₁ = [value], aₙ₊₁ = r·aₙ shows the 'multiply to get next term' pattern. For the bacteria: starts at 200, triples so r=3, after 1 hour: 2003=600, 2 hours:6003=1800, confirming geometric. With a₁=200, the explicit formula is aₙ=200·3^{n-1}. The recursive version: a₁=200, aₙ₊₁=3·aₙ. This captures 'start at 200, multiply by 3 each time.' Choice B is correct because it properly identifies the first term as a₁=200 and the common ratio as r=3, giving the recursive a₁=200, aₙ₊₁=3aₙ and explicit aₙ=200·3^{n-1}. Great work! Choice A confuses arithmetic with geometric: this sequence multiplies by the same amount each time, so it's geometric, not arithmetic. The pattern is multiplying by 3, not adding 3. Real-world clue: if the context mentions 'constant increase' or 'same amount each time,' think arithmetic. If it mentions 'percent growth,' 'doubles each,' or 'multiplies by,' think geometric. The language tells you which type! Writing explicit formulas: for arithmetic, use aₙ = a₁ + (n - 1)d (start with first term, add the difference (n-1) times); for geometric, use aₙ = a₁·r^(n-1) (start with first term, multiply by ratio (n-1) times). The (n-1) appears because the first term already includes one application!

Question 16

What is the explicit formula for the arithmetic sequence with $a_1=10$ and common difference $d=-3$ ?

$a_n=10-3(n-1)$ (correct answer)
$a_n=10-3n$
$a_n=10\cdot (-3)^{n-1}$
$a_n=-3+10(n-1)$

Explanation: This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Arithmetic sequences have a constant difference between consecutive terms: you add (or subtract) the same amount each time. The explicit formula aₙ = a₁ + (n - 1)d uses the first term a₁ and the common difference d to find any term directly, while the recursive form a₁ = [value], aₙ₊₁ = aₙ + d describes how to get each term from the one before. Given a₁ = 10 and d = -3, we apply the arithmetic explicit formula: aₙ = a₁ + (n-1)d = 10 + (n-1)·(-3) = 10 - 3(n-1). This gives us the sequence 10, 7, 4, 1, -2, ... (subtracting 3 each time). Choice A is correct because it properly uses the arithmetic explicit formula aₙ = a₁ + (n-1)d with a₁ = 10 and d = -3, giving aₙ = 10 - 3(n-1). Great work! Choice B uses the wrong formula structure: aₙ = 10 - 3n would give a₁ = 10 - 3(1) = 7, not 10. Remember, we need (n-1) in the formula, not just n! Writing explicit formulas: for arithmetic, use aₙ = a₁ + (n - 1)d (start with first term, add the difference (n-1) times); for geometric, use aₙ = a₁·r^(n-1) (start with first term, multiply by ratio (n-1) times). The (n-1) appears because the first term already includes one application!

Question 17

What is the common ratio of the geometric sequence $10,\ 5,\ 2.5,\ 1.25,\ \dots$ ?

$r=-2$
$r=2$
$r=\dfrac{1}{2}$ (correct answer)
$r=-\dfrac{1}{2}$

Explanation: This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Geometric sequences have a constant ratio between consecutive terms: you multiply by the same factor each time. The explicit formula aₙ = a₁ · r^(n-1) uses the first term a₁ and common ratio r, while the recursive form a₁ = [value], aₙ₊₁ = r·aₙ shows the 'multiply to get next term' pattern. To identify this as geometric and find the common ratio, we check the ratio between consecutive terms: 5/10 = 1/2, 2.5/5 = 1/2, 1.25/2.5 = 1/2, confirming common ratio r = 1/2. Each term is half of the previous term, so we multiply by 1/2 to get from one term to the next. Choice A is correct because it properly identifies the common ratio as r = 1/2, which means each term is half the previous term. Great work! Choice B has the reciprocal of the correct ratio: we divide by 2 (multiply by 1/2) to get from one term to the next, not multiply by 2. Check by comparing consecutive terms: 10 × (1/2) = 5, not 10 × 2 = 20! To identify arithmetic vs geometric: subtract consecutive terms (if differences are constant, it's arithmetic with d = that difference), or divide consecutive terms (if ratios are constant, it's geometric with r = that ratio). If neither pattern works, it's neither arithmetic nor geometric!

Question 18

You deposit money into an account: you start with $200 in week 1 and add $25 each week after that. Let $a_n$ be the amount in the account after $n$ weeks. Write both a recursive formula and an explicit formula for $a_n$ .

Recursive: $a_1=200,\ a_{n+1}=a_n\cdot 25$ ; Explicit: $a_n=200\cdot 25^{n-1}$
Recursive: $a_1=225,\ a_{n+1}=a_n+25$ ; Explicit: $a_n=225+25(n-1)$
Recursive: $a_1=200,\ a_{n+1}=a_n+25$ ; Explicit: $a_n=200+25(n-1)$ (correct answer)
Recursive: $a_1=200,\ a_{n+1}=a_n-25$ ; Explicit: $a_n=200-25(n-1)$

Explanation: This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Arithmetic sequences have a constant difference between consecutive terms: you add (or subtract) the same amount each time. The explicit formula aₙ = a₁ + (n - 1)d uses the first term a₁ and the common difference d to find any term directly, while the recursive form a₁ = [value], aₙ₊₁ = aₙ + d describes how to get each term from the one before. Looking at the context, we start with $200 in week 1 and add $25 each week, so we have: week 1: $200, week 2: $225, week 3: $250, etc. The common difference is d = 25. With first term a₁ = 200, the explicit formula is aₙ = 200 + (n - 1)·25. The recursive form is: a₁ = 200, aₙ₊₁ = aₙ + 25. Both formulas describe the pattern 'start at $200, add $25 each time'! Choice C is correct because it properly identifies the first term as a₁ = 200 and the common difference as d = 25, giving the recursive formula a₁ = 200, aₙ₊₁ = aₙ + 25 and explicit formula aₙ = 200 + 25(n-1). Great work! Choice B has the wrong first term: it uses a₁ = 225, which would be the amount after week 2, not week 1. Remember, a₁ represents the amount after week 1, which is the starting $200! Real-world clue: if the context mentions 'constant increase' or 'same amount each time,' think arithmetic. If it mentions 'percent growth,' 'doubles each,' or 'multiplies by,' think geometric. The language tells you which type!

Question 19

The sequence is defined recursively by $a_1=3$ and $a_{n+1}=a_n+7$ . What is the explicit formula for $a_n$ ?

$a_n=3+7n$
$a_n=3\cdot 7^{n-1}$
$a_n=7+3(n-1)$
$a_n=3+7(n-1)$ (correct answer)

Explanation: This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). The difference between recursive and explicit formulas: recursive is like climbing stairs one at a time (you need to know the previous term), while explicit is like taking an elevator directly to any floor (you can find the nth term without finding all the ones before it). Both describe the same sequence, just different approaches! Given the recursive formula a₁ = 3 and aₙ₊₁ = aₙ + 7, we identify: this is arithmetic (adding 7 each time) with first term a₁ = 3 and common difference d = 7. To convert to explicit form, we use aₙ = a₁ + (n-1)d = 3 + (n-1)·7 = 3 + 7(n-1). Choice C is correct because it properly uses the arithmetic explicit formula structure aₙ = a₁ + (n-1)d with a₁ = 3 and d = 7, giving aₙ = 3 + 7(n-1). Great work! Choice A has the right idea but uses the wrong formula structure: for arithmetic sequences, the explicit formula needs (n-1), not just n. The formula 3 + 7n would give a₁ = 10, not 3! Writing explicit formulas: for arithmetic, use aₙ = a₁ + (n - 1)d (start with first term, add the difference (n-1) times); for geometric, use aₙ = a₁·r^(n-1) (start with first term, multiply by ratio (n-1) times). The (n-1) appears because the first term already includes one application!

Question 20

What is the explicit formula for the arithmetic sequence with $a_1=3$ and common difference $d=4$ ?

$a_n = 3 + 4n$
$a_n = 3\cdot 4^{n-1}$
$a_n = 3 + 4(n-1)$ (correct answer)
$a_n = 4 + 3(n-1)$

Explanation: This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Arithmetic sequences have a constant difference between consecutive terms: you add (or subtract) the same amount each time. The explicit formula aₙ = a₁ + (n - 1)d uses the first term a₁ and the common difference d to find any term directly, while the recursive form a₁ = [value], aₙ₊₁ = aₙ + d describes how to get each term from the one before. Given that a₁ = 3 and d = 4, we can directly apply the explicit formula for arithmetic sequences: aₙ = a₁ + (n-1)d = 3 + (n-1)·4 = 3 + 4(n-1). This formula says 'start with 3, then add 4 exactly (n-1) times to get to the nth term.' Choice C is correct because it properly identifies the first term as a₁ = 3 and the common difference as d = 4, giving the explicit formula aₙ = 3 + 4(n-1). Great work! Choice A uses the wrong formula structure: for arithmetic sequences, the explicit formula needs aₙ = a₁ + d(n-1), but this choice has aₙ = 3 + 4n which would give a₁ = 3 + 4(1) = 7, not 3. Writing explicit formulas: for arithmetic, use aₙ = a₁ + (n - 1)d (start with first term, add the difference (n-1) times); for geometric, use aₙ = a₁·r^(n-1) (start with first term, multiply by ratio (n-1) times). The (n-1) appears because the first term already includes one application!