SOLUTION: Two terms of an arithmetic sequence are a7 = 33 and a22 = 138. Find a rule for the nth term.

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Question 388661: Two terms of an arithmetic sequence are a7 = 33 and a22 = 138. Find a rule for the nth term.
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
a%5B7%5D+=+33 and a%5B22%5D+=+138

a%5Bn%5D=a%5B1%5D%2B%28n-1%29d

Substitute n=7

a%5B7%5D=a%5B1%5D%2B%287-1%29d
a%5B7%5D=a%5B1%5D%2B6d

Substitute 33 for a%5B7%5D

33=a%5B1%5D%2B6d

----------------

a%5Bn%5D=a%5B1%5D%2B%28n-1%29d

Substitute n=22

a%5B22%5D=a%5B1%5D%2B%2822-1%29d
a%5B22%5D=a%5B1%5D%2B21d

Substitute 138 for a%5B22%5D

138=a%5B1%5D%2B21d

So we have this system of equations:

system%2833=a%5B1%5D%2B6d%2C+138=a%5B1%5D%2B21d%29

Solve that by substitution and get a%5B1%5D=-9 and d=7

So the rule is

a%5Bn%5D=a%5B1%5D%2B%28n-1%29d

a%5Bn%5D=-9%2B%28n-1%297

a%5Bn%5D=-9%2B7%28n-1%29

a%5Bn%5D=-9%2B7n-7

a%5Bn%5D=7n-16

To check substitute n = 7 and n = 22 and see if we get 33 and 138

a%5B7%5D=7%287%29-16
a%5B7%5D=49-16
a%5B7%5D=33

a%5B22%5D=7%2822%29-16
a%5B7%5D=154-16
a%5B7%5D=138

It checks, so the answer is

a%5Bn%5D=7n-16

Edwin