# Algebra 1 : How to find the nth term of an arithmetic sequence

## Example Questions

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### Example Question #1 : 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 .

For us, is and is .

Now we can solve for .

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

### Example Question #1 : How To Find The Nth Term Of An Arithmetic Sequence

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 .

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 .

### Example Question #1 : How To Find The Nth Term Of An Arithmetic Sequence

An arithmetic sequence begins with . If  is the first term in the sequence, find the 31st term.

Explanation:

For arithmetic sequences, we use the formula , where  is the term we are trying to find,  is the first term, and  is the difference between consecutive terms. In this case,  and . So, we can write the formula as , and .

### Example Question #1 : How To Find The Nth Term Of An Arithmetic Sequence

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 , or, equivalently,

, or equivalently,

### Example Question #5 : How To Find The Nth Term Of An Arithmetic Sequence

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:

### Example Question #6 : How To Find The Nth Term Of An Arithmetic Sequence

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 :

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

### Example Question #1 : How To Find The Nth Term Of An Arithmetic Sequence

Find the 100th term in the following arithmetic sequence

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, .  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

### Example Question #1 : How To Find The Nth Term Of An Arithmetic Sequence

To find any term of an arithmetic sequence:

Where  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.

Explanation:

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:

### Example Question #9 : How To Find The Nth Term Of An Arithmetic Sequence

To find any term of an arithmetic sequence:

Where  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.

Explanation:

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.

### Example Question #1 : How To Find The Nth Term Of An Arithmetic Sequence

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

1, 5, 9, 13, . . .

37

45

53

41

49

41

Explanation:

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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