There are many problems we can solve if we keep in mind that the nth term of an arithmetic sequence can be written in the following way:

a_n = a₁+(n - 1)d

Where a₁ is the first term, and d is the common difference.

For example, if we are told that the first two terms add up to the fifth term, and that the common difference is 8 less than the first term we can take this equation:

a₁ + a₂ = a₅

and rewrite it as follows:

a₁ + [a₁ + d] = [a₁ + 4d]

This leads to a₁ = 3d.

Combine this with d = a₁ - 8, and we have:

a₁ = 3(a₁ - 8)

or a₁ = 12.

This leads to d = 4, and from this information, we can find any other term of the sequence.

1.

The first term and the second term of an arithmetic sequence add to 15. The second and third term add to 21. What is the common difference?

2.

The first three terms of an arithmetic sequence add to 63 less than the sum of the fourth, fifth, and sixth terms. What is the common difference?

3.

The sum of the first two terms of an arithmetic sequence is 20 less than the third term. What is the common difference, in terms of the first term?

4.

If the third term of an arithmetic sequence is 14, and the fifth term is 22, what is the eighth term?

5.

The sum of the first four terms of an arithmetic sequence is 100, and the common difference is 2. What is the first term?

6.

In an arithmetic sequence, the common difference is twice the first term, and the second term is 120. What is the third term?

7.

In an arithmetic sequence, all the terms are odd. If the first term is less than 10, and the second term is greater than 10, what is the least possible value of the 4th term?

8.

The sum of the first two terms of an arithmetic sequence is equal to the sum of the third and fourth terms. What is the common difference?

9.

The common difference in an arithmetic sequence is -1. If the sum of the first 4 terms is 62, what is the first term?

10.

The sum of the first 2 terms of an arithmetic sequence is 4 more than the sum of the first 2 terms of an arithmetic sequence with the same first term, but the opposite common difference. What is the common difference?