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

 Mar 4, 2021
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difference d=−31/9.
We have that a5=9
This means that  +d(5−1)=9 or  a1+4d=9.
We have that a32=−84.
This means that a1+d(32−1)=−84 or a1+31d=−84.
We have that d=−319.
We obtained the following system:
a1+4d=9
a1+31d=−84
d=−31/9
Solving it, we obtain that a1=205/9, d=-31/9.
Finally,
a23=a1+d(23−1)=205/9+(−31/9)(23−1)=−53.

 Mar 4, 2021

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