Solving for the 19th Term in an Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant, the common difference d, to the preceding term. Given that the 13th term and the 7th term of an arithmetic sequence are 15 and 51 respectively, this article will guide you through the process of finding the term that equals -21 using the common difference and the first term.

In an arithmetic sequence, each term can be expressed as:

a_n a_1 n-1d

Identifying Given Information

First, let's identify the given information:

The 13th term, a_{13}, is 15. The 7th term, a_{7}, is 51. We need to find the term that equals -21.

Step 1: Determining the Common Difference d

To find the common difference, we can set up the following equations based on the given information:

a_{13} a_1 12d 15

a_{7} a_1 6d 51

Subtract the second equation from the first:

(a_1 12d) - (a_1 6d) 15 - 51

6d -36

d -6

Step 2: Finding the First Term a_1

Now that we have the common difference, we can substitute it back into one of the original equations to find the first term a_1:

a_1 6d 51

a_1 6(-6) 51

a_1 - 36 51

a_1 87

Step 3: Finding the Term that Equals -21

Now that we have the first term and the common difference, we can find the term that equals -21 using the formula:

a_n a_1 (n-1)d

-21 87 (n-1)(-6)

-21 87 - 6n 6

-21 - 87 - 6 -6n

-114 -6n

n 19

Therefore, the term that equals -21 is the 19th term of the sequence.

Verification:

T_{19} 87 (19-1)(-6)

T_{19} 87 - 108 -21

Summary

In this article, we have demonstrated how to find the term of an arithmetic sequence when given the values of specific terms. By following the steps of identifying the given information, determining the common difference, finding the first term, and then using the formula for the arithmetic sequence, we were able to find that the 19th term of the sequence is -21.