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

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

Explanation

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.

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

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

It cannot be determined from the information given.

Explanation

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 .

What is the 11th term in the following sequence?

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

Explanation

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.

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

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

Explanation

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 $a_{31} = 141$ .

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

Explanation

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,

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

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

Explanation

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:

$a + 7\cdot 6= 87$

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

Explanation

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:

$a + 8 \cdot12 = 87$

Find the 100th term in the following arithmetic sequence

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

Explanation

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

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

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

$x_{100}=4+693$

$x_{100}=697$

What is the 20th term of ?

$\textup{Cannot be determined.}$

Explanation

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.