College Algebra Quiz: Arithmetic Sequences Explicit And Recursive Forms

8 questions · exam conditions

0:00

Arithmetic Sequences Explicit And Recursive FormsQuestion 1 of 8

An arithmetic sequence has its 4th term equal to 22 and its 10th term equal to 46. What is the explicit formula for the $n$ th term of this sequence?

a_n = 4 + 4(n-1)

a_n = 6 + 4(n-1)

a_n = 10 + 4(n-1)

a_n = 22 + 4(n-4)

← Back to quizzes

College Algebra Quiz

College Algebra Quiz: Arithmetic Sequences Explicit And Recursive Forms

Practice Arithmetic Sequences Explicit And Recursive Forms in College 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 Sequences Explicit And Recursive Forms, giving you a quick way to practice the rules, question types, and explanations that matter most for College 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

An arithmetic sequence has its 4th term equal to 22 and its 10th term equal to 46. What is the explicit formula for the $n$ th term of this sequence?

$a_n = 4 + 4(n-1)$
$a_n = 6 + 4(n-1)$
$a_n = 10 + 4(n-1)$ (correct answer)
$a_n = 22 + 4(n-4)$

Explanation: When you encounter an arithmetic sequence problem with two given terms, you need to find the common difference first, then use it to determine the explicit formula. In an arithmetic sequence, the difference between consecutive terms is constant. Given that

a_4 = 22

and

a_{10} = 46

, you can find the common difference by recognizing that from the 4th term to the 10th term, there are 6 steps (since 10 - 4 = 6). The total change is 46 - 22 = 24, so the common difference is

d = \frac{24}{6} = 4

. Now you can work backwards to find the first term. Since

a_4 = 22

and each term decreases by 4 as you go backwards:

a_3 = 18

a_2 = 14

, and

a_1 = 10

. The explicit formula is

a_n = 10 + 4(n-1)

, which is choice C. Let's check why the other options fail. Choice A gives

a_1 = 4

, which would make

a_4 = 4 + 4(3) = 16

, not 22. Choice B gives

a_1 = 6

, making

a_4 = 6 + 4(3) = 18

, still incorrect. Choice D uses a different reference point but gives

a_1 = 22 + 4(-3) = 10

, which seems right, but when you check

a_{10}

, you get

22 + 4(6) = 46

—this works too! However, D isn't in standard form. Study tip: Always verify your formula by plugging in the given term positions. The standard explicit formula form

a_n = a_1 + d(n-1)

is usually expected, so convert any alternative forms to this structure.

Question 2

A sequence is defined recursively as $a_1 = -3$ and $a_n = a_{n-1} + 4$ for $n \geq 2$ . Which explicit formula correctly represents the 12th term of this sequence?

$a_{12} = -3 + 12 \cdot 4 = 45$
$a_{12} = -3 + 11 \cdot 4 = 41$ (correct answer)
$a_{12} = -3 + 4^{12} = 16{,}777{,}213$
$a_{12} = 12(-3 + 4) = 12$

Explanation: For an arithmetic sequence with

a_1 = -3

and common difference

d = 4

, the explicit formula is

a_n = a_1 + (n-1)d

. For the 12th term:

a_{12} = -3 + (12-1) \cdot 4 = -3 + 11 \cdot 4 = 41

. Choice A incorrectly uses

n

instead of

(n-1)

. Choice C treats it as exponential instead of arithmetic. Choice D incorrectly applies the formula structure.

Question 3

A student writes two forms for the same arithmetic sequence: Explicit form: $a_n = 8 + 3(n-1)$ and Recursive form: $a_1 = k$ , $a_n = a_{n-1} + d$ for $n \geq 2$ . If both forms represent the same sequence, what are the values of $k$ and $d$ ?

$k = 3, d = 8$
$k = 5, d = 3$
$k = 8, d = 4$
$k = 8, d = 3$ (correct answer)

Explanation: When you encounter questions comparing explicit and recursive forms of arithmetic sequences, you need to identify the first term and common difference from each form, then match them up. The explicit form

a_n = 8 + 3(n-1)

follows the standard pattern

a_n = a_1 + d(n-1)

, where

a_1

is the first term and

d

is the common difference. Comparing these, you can see that

a_1 = 8

and

d = 3

. You can verify this by finding

a_1 = 8 + 3(1-1) = 8 + 0 = 8

. The recursive form states

a_1 = k

and

a_n = a_{n-1} + d

. Since both forms represent the same sequence, the first term

k

must equal 8, and the common difference

d

must equal 3. Looking at the wrong answers: Choice A switches the values, giving

k = 3

and

d = 8

. This likely comes from misreading the explicit form's structure. Choice B gives

k = 5

, which might result from incorrectly calculating

a_1

8 + 3 - 1 = 10

or some other computational error, but keeps the correct common difference. Choice C correctly identifies

k = 8

but incorrectly states

d = 4

, perhaps by misreading the coefficient in the explicit form. Therefore,

k = 8

and

d = 3

, making D correct. Study tip: Always extract the first term and common difference systematically from explicit forms using the pattern

a_n = a_1 + d(n-1)

, and double-check by computing

a_1

directly when

n = 1

Question 4

The explicit formula for an arithmetic sequence is $a_n = 23 - 5(n-1)$ . What are the first three terms of this sequence, and what is the recursive definition?

First three terms: 23, 18, 13; Recursive: $a_1 = 23$ , $a_n = a_{n-1} - 5$ (correct answer)
First three terms: 18, 13, 8; Recursive: $a_1 = 18$ , $a_n = a_{n-1} - 5$
First three terms: 23, 28, 33; Recursive: $a_1 = 23$ , $a_n = a_{n-1} + 5$
First three terms: 23, 18, 13; Recursive: $a_1 = 23$ , $a_n = a_{n-1} + 5$

Explanation: From

a_n = 23 - 5(n-1)

a_1 = 23 - 5(0) = 23

a_2 = 23 - 5(1) = 18

a_3 = 23 - 5(2) = 13

. The common difference is

d = -5

, so the recursive formula is

a_1 = 23

and

a_n = a_{n-1} - 5

. Choice B incorrectly starts with

a_2

. Choice C and D incorrectly use positive 5 as the common difference.

Question 5

An arithmetic sequence starts with $a_1 = 15$ and has a common difference of $d = -3$ . For which value of $n$ does the explicit formula $a_n = 15 - 3(n-1)$ first give a negative result?

$n = 5$
$n = 6$
$n = 7$ (correct answer)
$n = 8$

Explanation: Set

a_n < 0

15 - 3(n-1) < 0

. This gives

15 < 3(n-1)

, so

5 < n-1

, which means

n > 6

. The first integer value where

n > 6

n = 7

. Checking:

a_6 = 15 - 3(5) = 0

and

a_7 = 15 - 3(6) = -3

. Choice A gives

a_5 = 3 > 0

. Choice B gives

a_6 = 0

. Choice D is too late since

a_7

is already negative.

Question 6

An arithmetic sequence has $a_1 = 7$ and $a_{15} = 91$ . What is the recursive formula for this sequence?

$a_1 = 7$ ; $a_n = a_{n-1} + 6$ for $n \geq 2$ (correct answer)
$a_1 = 7$ ; $a_n = a_{n-1} + 7$ for $n \geq 2$
$a_1 = 7$ ; $a_n = a_{n-1} + 14$ for $n \geq 2$
$a_1 = 7$ ; $a_n = a_{n-1} + 84$ for $n \geq 2$

Explanation: To find the common difference, use

a_n = a_1 + (n-1)d

. With

a_{15} = 91

and

a_1 = 7

91 = 7 + (15-1)d

, so

84 = 14d

, giving

d = 6

. The recursive formula is

a_1 = 7

and

a_n = a_{n-1} + 6

for

n \geq 2

. Choice B uses the first term as the common difference. Choice C uses the number of steps between terms. Choice D uses the total difference between the given terms.

Question 7

Consider the arithmetic sequence defined by $a_n = 5n - 2$ . Which of the following correctly expresses this sequence in recursive form?

$a_1 = 3$ ; $a_n = a_{n-1} + 2$ for $n \geq 2$
$a_1 = 5$ ; $a_n = a_{n-1} - 2$ for $n \geq 2$
$a_1 = -2$ ; $a_n = a_{n-1} + 5$ for $n \geq 2$
$a_1 = 3$ ; $a_n = a_{n-1} + 5$ for $n \geq 2$ (correct answer)

Explanation: When converting between explicit and recursive forms of arithmetic sequences, you need to identify two key components: the first term and the common difference (the amount added between consecutive terms). Let's work with the given explicit formula

a_n = 5n - 2

. First, find the first term by substituting

n = 1

a_1 = 5(1) - 2 = 3

. Next, determine the common difference by calculating a few terms. We have

a_1 = 3

a_2 = 5(2) - 2 = 8

, and

a_3 = 5(3) - 2 = 13

. The difference between consecutive terms is

8 - 3 = 5

and

13 - 8 = 5

, so the common difference is 5. The recursive form states that each term equals the previous term plus the common difference:

a_1 = 3

and

a_n = a_{n-1} + 5

for

n \geq 2

. This matches option D. Option A has the correct first term but uses a common difference of 2 instead of 5. Option B incorrectly calculates

a_1 = 5

(this would be

a_2

) and subtracts 2, which would create a decreasing sequence. Option C uses

a_1 = -2

, which would be

a_0

if we extended the sequence backward, and while it has the correct common difference of 5, the wrong starting point makes the entire recursive definition incorrect. Study tip: Always verify your recursive form by computing the first few terms and checking they match the explicit formula. The coefficient of

n

in an arithmetic sequence's explicit form is always the common difference.

Question 8

An arithmetic sequence has the property that $a_5 = 17$ and $a_9 = 29$ . If this sequence is defined recursively starting with $a_1$ , what is the value of $a_1$ ?

$a_1 = 2$
$a_1 = 5$ (correct answer)
$a_1 = 8$
$a_1 = 11$

Explanation: First find the common difference:

d = \frac{a_9 - a_5}{9 - 5} = \frac{29 - 17}{4} = 3

. Using

a_5 = a_1 + 4d

17 = a_1 + 4(3) = a_1 + 12

, so

a_1 = 5

. Choice A results from using the wrong relationship between terms. Choice C comes from confusing the position number with the term value. Choice D results from incorrectly calculating the common difference.