How to find the nth term of an arithmetic sequence - Algebra

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Question

Given the the sequence below, what is the 11th term of the sequence?

1, 5, 9, 13, . . .

Answer

The 11th term means there are 10 gaps in between the first term and the 11th term. Each gap has a difference of +4, so the 11th term would be given by 10 * 4 + 1 = 41.

The first term is 1.

Each term after increases by +4.

The nth term will be equal to 1 + (n – 1)(4).

The 11th term will be 1 + (11 – 1)(4)

1 + (10)(4) = 1 + (40) = 41

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Question

Consider the following arithmetic sequence:

What is the term?

Answer

A simple way to find the term of an arithmetic sequence is to use the formula $a_{n} = a_{1} + (d)(n-1)$ .

Here, $a_{n}$ is the term you are trying to find, $a_{1}$ is the first term, and is the common difference. For this question, the common difference is .

$a_{n} = 3 + (11)(37-1)$

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Question

The second term of an arithmetic sequence is ; the fourth term is . What is the first term?

Answer

The common difference between the terms is half that between the second and fourth terms - that is:

$\small \frac{(10x - 32) - 4x}{2} = \frac{6x - 32 }{2} = 3x - 16$

Subtract this common difference from the second term to get the first:

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Question

An arithmetic sequence is given by the formula $a_{n} = 4 + 3 n$ . What is the difference between $a_{13}$ and $a_{10}$

Answer

You can either calculate the vaules of $a_{13}$ and $a_{10}$ and subtract, or notice from the formula that each succesive number in the sequence is 3 larger than the previous

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Question

The first term of an arithmetic sequence is ; the fifth term is . What is the second term?

Answer

To find the common difference , use the formula $a_{1}+4d=a_{5}$ .

For us, is and is .

Now we can solve for .

Add the common difference to the first term to get the second term.

$a_{2}=a_{1}+d=10+7 = 17$

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Question

The sum of the first three terms of an arithmetic sequence is 111 and the fourth term is 49. What is the first term?

Answer

Let be the common difference, and let be the second term. The first three terms are, in order, .

The sum of the first three terms is .

$x=\frac{111}{3}=37$

Now we know that the second term is 37. The fourth term is the second term plus twice the common difference: . Since the second and fourth terms are 37 and 49, respectively, we can solve for the common difference.

The common difference is 6. The first term is .

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Question

What is the 11th term in the following sequence?

$\left { 87, 78, 69, 60...\right }$

Answer

First we need to show how this sequence is changing. Let's call the first number $n_{1}$ , and the second number $n_{2}$ , and so on.

$n_{1}-n_{2}=9$

$n_{2}-n_{3}=9$

Ok, so we have established that the sequence is shrinking by 9 each time. So now we need to calculate out to the 11th term. Starting from the first term, we need to subtract 9 ten times to get to the 11th term. So that would look like this.

$n_{11}=n_{1}+(10\times -9)=87+(10\times -9)=-3$

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Question

An arithmetic sequence begins with $\left { -9,-4,1,6...\right }$ . If is the first term in the sequence, find the 31st term.

Answer

For arithmetic sequences, we use the formula $a_{n} = a_{1}+(n-1)d$ , where $a_{n}$ is the term we are trying to find, $a_{1}$ is the first term, and is the difference between consecutive terms. In this case, $a_{1} = -9$ and . So, we can write the formula as $a_{31} = -9 +(30)(5)$ , and $a_{31} = 141$ .

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Question

The fourth and tenth terms of an arithmetic sequence are 372 and 888, respectively. What is the first term?

Answer

Let be the common difference of the sequence. Then $a_{4} + 6d = a_{10}$ , or, equivalently,

$d = \frac{a_{10} - a_{4}}{6}= \frac{888 - 372}{6} = \frac{516}{6}= 86$

$a_{1} + 3d = a_{4 }$ , or equivalently,

$a_{1} = a_{4 }-3d = 372-3\cdot 86 = 114$

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Question

The ninth and tenth terms of an arithmetic sequence are, respectively, 87 and 99. What is its first term?

Answer

The common difference of the sequence is the difference of the tenth and ninth terms: .

The ninth term of an arithmetic sequence with first term and common difference is , so we set this equal to 87, set , and solve:

$a + 8 \cdot12 = 87$

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Question

The eighth and tenth terms of an arithmetic sequence are, respectively, 87 and 99. What is its first term?

Answer

The eighth and tenth terms of the sequence are and , where is the first term and is the common difference. We can find the common difference by subtracting the tenth and eighth terms and solving for :

$\underline{a+7d = 87}$

$2d \div 2= 12 \div 2$

Now set eighth term equal to 87, set , and solve:

$a + 7\cdot 6= 87$

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Question

Find the 100th term in the following arithmetic sequence

$4,\ 11,\18,\25,\32,\39,\46,...$

Answer

Before we can figure out the 100th term, we need to find a rule for this arithmetic sequence. Remember, the general rule for this sequence is

where represents the first number in the sequence, is the common difference between consecutive numbers, and is the -th number in the sequence.

In our problem, . Also, each time we move up from one number to another, the number increases by 7. Therefore, $\ d=7$ . So the rule for this sequence is written as

Now that we found our rule, we can go on and figure out what the 100th term is equal to. For the 100th term, . Thus

$x_{100}=4+7(100-1)$

$x_{100}=4+7(99)$

$x_{100}=4+693$

$x_{100}=697$

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Question

To find any term of an arithmetic sequence:

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

Where $a_{1}$ is the first term, is the number of the term to find, and is the common difference in the sequence.

Find the 18th term of the following arithmetic sequence.

$\left { 2,5,8,11,14...\right }$

Answer

Start by finding the common difference, , in this sequence, which you can get by subtracting the first term from the second.

Then, using the formula given before the question:

$a_{18}=2+(18-1)3=2+(17)3=2+51=53$

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Question

To find any term of an arithmetic sequence:

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

Where $a_{1}$ is the first term, is the number of the term to find, and is the common difference in the sequence.

Find the 26th term of the following arithmetic sequence.

$\left { 101,92,83...\right }$

Answer

Start by finding the common difference in terms by subtracting the first term from the second.

Then, fill in the rest of the equation given before the question.

$a_{26}=101+(26-1)-9=101+(25)-9$

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Question

What is the 20th term of ?

Answer

Notice that every odd term is increasing at an increment of , since the numbers are . We will ignore these terms and concentrate on only the even terms.

The even terms are for the second, fourth, and sixth terms respectively, which means that the numbers are a multiple of nine.

The equation that represents the terms for this sequence is: $y=\frac{9}{2}n$ , for even terms since it only applies to the even terms that are divisible by nine and occurs after every two terms.

Substitute to determine the 20th term.

$y=\frac{9}{2}(20) = 9(10) =90$

The answer is:

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Question

Given the the sequence below, what is the 11th term of the sequence?

1, 5, 9, 13, . . .

Answer

The 11th term means there are 10 gaps in between the first term and the 11th term. Each gap has a difference of +4, so the 11th term would be given by 10 * 4 + 1 = 41.

The first term is 1.

Each term after increases by +4.

The nth term will be equal to 1 + (n – 1)(4).

The 11th term will be 1 + (11 – 1)(4)

1 + (10)(4) = 1 + (40) = 41

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Question

Consider the following arithmetic sequence:

What is the term?

Answer

A simple way to find the term of an arithmetic sequence is to use the formula $a_{n} = a_{1} + (d)(n-1)$ .

Here, $a_{n}$ is the term you are trying to find, $a_{1}$ is the first term, and is the common difference. For this question, the common difference is .

$a_{n} = 3 + (11)(37-1)$

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Question

The second term of an arithmetic sequence is ; the fourth term is . What is the first term?

Answer

The common difference between the terms is half that between the second and fourth terms - that is:

$\small \frac{(10x - 32) - 4x}{2} = \frac{6x - 32 }{2} = 3x - 16$

Subtract this common difference from the second term to get the first:

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Question

An arithmetic sequence is given by the formula $a_{n} = 4 + 3 n$ . What is the difference between $a_{13}$ and $a_{10}$

Answer

You can either calculate the vaules of $a_{13}$ and $a_{10}$ and subtract, or notice from the formula that each succesive number in the sequence is 3 larger than the previous

Compare your answer with the correct one above

Question

The first term of an arithmetic sequence is ; the fifth term is . What is the second term?

Answer

To find the common difference , use the formula $a_{1}+4d=a_{5}$ .

For us, is and is .

Now we can solve for .

Add the common difference to the first term to get the second term.

$a_{2}=a_{1}+d=10+7 = 17$

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