How to find the next term in an arithmetic sequence - SAT Math

Card 0 of 88

Question

A sequence of numbers is as follows:

What is the sum of the first seven numbers in the sequence?

Answer

The pattern of the sequence is (x+1) * 2.

We have the first 5 terms, so we need terms 6 and 7:

(78+1) * 2 = 158

(158+1) * 2 = 318

3 + 8 + 18 +38 + 78 + 158 + 318 = 621

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Question

What is the next number in the following series: 0, 3, 8, 15, 24 . . . ?

Answer

The series is defined by n2 – 1 starting at n = 1. The sixth number in the series then equal to 62 – 1 = 35.

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Question

$\small 2, 3, 5, 9, 17, x, 65, 129....$

$\small \textup{In the above number sequence, what is the value of},x?$

Answer

Each term in the sequence is one less than twice the previous term.

So, $\small x=2\left ( 17\right )-1$

$\small x=33$

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Question

Find the next term of the following sequence:

Answer

The sequence provided is arithmetic. An arithmetic sequence has a common difference between each consecutive term. In this case, the difference is ; therefore, the next term is .

You can also use a formula to find the next term of an arithmetic sequence:

$a_{n}=a_{n-1}+d$

where the current term and the common difference.

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Question

Solve each problem and decide which is the best of the choices given.

Find the sixth term in the following arithmetic sequence.
$\frac{1}{4},\frac{1}{8},0...$

Answer

First find the common difference of the sequence,

$\frac{1}{4}-\frac{1}{8}=\frac{1}{8}$

$\frac{1}{8}-0=\frac{1}{8}$

Thus there is a common difference of

$\frac{1}{8}$ between each term,

so follow that pattern for another terms, and the result is $-\frac{3}{8}$ .

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Question

Find the missing number in the sequence:

Answer

The pattern of this sequence is where represents the position of the number in the sequence.

for the first number in the sequence.

for the second number.

For the fourth term, . Therefore, .

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Question

Complete the sequence:

Answer

The pattern of this sequence is where represents the place of each number in the order of the sequence.

Here are our givens:

, our first term.

, our second term.

, our third term.

This means that our fourth term will be:

.

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Question

An arithmetic sequence begins as follows:

$\sqrt{20} , \sqrt{180},...$

Express the next term of the sequence in simplest radical form.

Answer

Using the Product of Radicals principle, we can simplify the first two terms of the sequence as follows:

$a_{1}=\sqrt{20} = \sqrt{4} \cdot \sqrt{5} = 2 \sqrt{5}$

$a_{2}= \sqrt{180} = \sqrt{36} \cdot \sqrt{5} = 6 \sqrt{5}$

The common difference of an arithmetic sequence can be found by subtracting the first term from the second:

$d = a_{2} - a_{1}$

$d = 6 \sqrt{5} - 2\sqrt{5} = ( 6 - 2 )\sqrt{5} = 4 \sqrt{5}$

Add this to the second term to obtain the desired third term:

$a_{3} = a_{2} +d = 6 \sqrt{5} + 4 \sqrt{5} =( 6 + 4) \sqrt{5} = 10 \sqrt{5}$ .

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Question

An arithmetic sequence begins as follows:

Give the sixth term of the sequence.

Answer

The common difference of an arithmetic sequence can be found by subtracting the first term from the second:

$d = a_{2} - a_{1}$

Setting $a_{1}= 4, a_{2}= 5i$ :

The th term $a_{n}$ of an arithmetic sequence can be derived using the formula

$a_{n} = a_{1} + (n-1)d$

Setting $a_{1}= 4, n = 6, d= -4+5i$

$a_{6} =4 + (6-1) ( -4+5i)$

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Question

An arithmetic sequence begins as follows:

$\sqrt{5} , \sqrt{10},...$

Give the next term of the sequence in simplest radical form.

Answer

Since no perfect square integer greater than 1 divides evenly into 5 or 10, both of the first two terms of the sequence are in simplest form.

The common difference of an arithmetic sequence can be found by subtracting the first term from the second:

$d = a_{2} - a_{1}$

Setting $a_{1}= \sqrt{5}, a_{2}= \sqrt{10}$ :

$d = \sqrt{10} - \sqrt{5}$

Add this to the second term to obtain the desired third term:

$a_{3} = a_{2} +d =\sqrt{10} + ( \sqrt{10} - \sqrt{5}) = \sqrt{10} + \sqrt{10} - \sqrt{5} = 2\sqrt{10}- \sqrt{5}$

This is not among the given choices.

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Question

An arithmetic sequence begins as follows:

$\frac{1}{5} , \frac{1}{3} , ...$

Give the sixth term of the sequence in decimal form.

Answer

The common difference of an arithmetic sequence can be found by subtracting the first term from the second:

$d = a_{2} - a_{1}$

Setting $a_{1}= \frac{1}{5}, a_{2}= \frac{1}{3}$ :

$d = \frac{1}{3} - \frac{1}{5} = \frac{5}{15} - \frac{3}{15} = \frac{2}{15}$

The th term $a_{n}$ of an arithmetic sequence can be derived using the formula

$a_{n} = a_{1} + (n-1)d$

Setting $n = 6 , a_{1} = \frac{1}{5}= \frac{3}{15}, d = \frac{2}{15}$ :

$a_{n} = \frac{3}{15} + (6-1) \cdot \frac{2}{15}$

$= \frac{3}{15} +5 \cdot \frac{2}{15}$

$= \frac{3}{15} + \frac{10}{15}$

$= \frac{13}{15}$

The decimal equivalent of this can be found by dividing 13 by 15 as follows:

The correct choice is $0.8\overline{6}$ .

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Question

A sequence of numbers is as follows:

What is the sum of the first seven numbers in the sequence?

Answer

The pattern of the sequence is (x+1) * 2.

We have the first 5 terms, so we need terms 6 and 7:

(78+1) * 2 = 158

(158+1) * 2 = 318

3 + 8 + 18 +38 + 78 + 158 + 318 = 621

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Question

What is the next number in the following series: 0, 3, 8, 15, 24 . . . ?

Answer

The series is defined by n2 – 1 starting at n = 1. The sixth number in the series then equal to 62 – 1 = 35.

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Question

$\small 2, 3, 5, 9, 17, x, 65, 129....$

$\small \textup{In the above number sequence, what is the value of},x?$

Answer

Each term in the sequence is one less than twice the previous term.

So, $\small x=2\left ( 17\right )-1$

$\small x=33$

Compare your answer with the correct one above

Question

Find the next term of the following sequence:

Answer

The sequence provided is arithmetic. An arithmetic sequence has a common difference between each consecutive term. In this case, the difference is ; therefore, the next term is .

You can also use a formula to find the next term of an arithmetic sequence:

$a_{n}=a_{n-1}+d$

where the current term and the common difference.

Compare your answer with the correct one above

Question

Solve each problem and decide which is the best of the choices given.

Find the sixth term in the following arithmetic sequence.
$\frac{1}{4},\frac{1}{8},0...$

Answer

First find the common difference of the sequence,

$\frac{1}{4}-\frac{1}{8}=\frac{1}{8}$

$\frac{1}{8}-0=\frac{1}{8}$

Thus there is a common difference of

$\frac{1}{8}$ between each term,

so follow that pattern for another terms, and the result is $-\frac{3}{8}$ .

Compare your answer with the correct one above

Question

Find the missing number in the sequence:

Answer

The pattern of this sequence is where represents the position of the number in the sequence.

for the first number in the sequence.

for the second number.

For the fourth term, . Therefore, .

Compare your answer with the correct one above

Question

Complete the sequence:

Answer

The pattern of this sequence is where represents the place of each number in the order of the sequence.

Here are our givens:

, our first term.

, our second term.

, our third term.

This means that our fourth term will be:

.

Compare your answer with the correct one above

Question

An arithmetic sequence begins as follows:

$\sqrt{20} , \sqrt{180},...$

Express the next term of the sequence in simplest radical form.

Answer

Using the Product of Radicals principle, we can simplify the first two terms of the sequence as follows:

$a_{1}=\sqrt{20} = \sqrt{4} \cdot \sqrt{5} = 2 \sqrt{5}$

$a_{2}= \sqrt{180} = \sqrt{36} \cdot \sqrt{5} = 6 \sqrt{5}$

The common difference of an arithmetic sequence can be found by subtracting the first term from the second:

$d = a_{2} - a_{1}$

$d = 6 \sqrt{5} - 2\sqrt{5} = ( 6 - 2 )\sqrt{5} = 4 \sqrt{5}$

Add this to the second term to obtain the desired third term:

$a_{3} = a_{2} +d = 6 \sqrt{5} + 4 \sqrt{5} =( 6 + 4) \sqrt{5} = 10 \sqrt{5}$ .

Compare your answer with the correct one above

Question

An arithmetic sequence begins as follows:

Give the sixth term of the sequence.

Answer

The common difference of an arithmetic sequence can be found by subtracting the first term from the second:

$d = a_{2} - a_{1}$

Setting $a_{1}= 4, a_{2}= 5i$ :

The th term $a_{n}$ of an arithmetic sequence can be derived using the formula

$a_{n} = a_{1} + (n-1)d$

Setting $a_{1}= 4, n = 6, d= -4+5i$

$a_{6} =4 + (6-1) ( -4+5i)$

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