SOLUTION: The fifth term of an arithmetic sequence is 9 and the 32nd term is -84. What is the 23rd term?

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Question 979756: The fifth term of an arithmetic sequence is 9 and the 32nd term is -84. What is the 23rd term?
Answer by ikleyn(52790) About Me  (Show Source):
You can put this solution on YOUR website!

a%5B5%5D = a%5B1%5D + 4%2Ad = 9,

a%5B32%5D = a%5B1%5D + 31%2Ad = -84.

Now distract the first equation from the second one. You will get

%2831+-+4%29%2Ad = -84 - 9,

27%2Ad = -93.

Thus  d = -93%2F27.
This is the  common difference  of the given arithmetic progression.

Next,   a%5B23%5D = a%5B1%5D + 22%2Ad = a%5B1%5D+%2B+4%2Ad + %2822-4%29%2Ad = a%5B5%5D + 18%2Ad = a%5B5%5D + 18%2A%28-93%2F27%29 = 9 - 2%2A%2893%2F3%29 = 9 - 2%2A31 = 9 - 62 = -53.

Answer.  a%5B23%5D = -53.