Question

In an arithmetic sequence, the 4th term is -66 and the 18th term is 172.

Determine the 12th term.

Answer 1

12th term = 70

Explanation:

The formula for finding the nth term of an arithmetic sequence is

`a_(n)=a_(1)+(n-1)d`, where an is the nth term, a1 is the first term, n is the term position (e.g., 12th or 18th), and d is the common difference.

Before we can find the 12th term, we will need to find both the first term and the common difference.

Because we already know the 4th term and 18th term, we can use a system of equations to find first the initial term, then the common difference:

`-66=a_(1)+(4-1)d\\ 172=a_(1)+(18-1)d\\ \\-66=a_(1)+3d\\ 172=a_(1)+17d`

We can eliminate d to find a1:

`-17(-66=a_(1)+3d)\\3(172=a_(1)+17d)\\ \\1122=-17a_(1) -51d\\516=3a_(1) +51d\\\\1638=-14a_(1)\\ -117=a_(1)`

Since we now know that -117 is the first term, we can simply plug in -66 for an, 4 for n, and -117 for a1 to find d (the common difference):

`-66=-117(4-1)d\\-66=-117+3d\\51=3d\\17=d`

Since we know have both the first term and the common difference, we plug these both in and 12 for n to find the 12th term:

`a_(12)=-117+(12-1)*17\\a_(12)=-117+11*17\\a_(12)=-117+187\\ a_(12)=70`