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If 9th, 13th and 15th terms of an arithmetical progression are the first three terms of a geometric series whose sum of infinite terms is 80, then find the first term for A.P and also for G.P and also the common difference and also the common ratio

If 9th, 13th and 15th terms of an arithmetical progression are the first three terms of a geometric series whose sum of infinite terms is 80, then find the first term for A.P and also for G.P and also the common difference and also the common ratio

Grade:11

1 Answers

Sunil Kumar FP
askIITians Faculty 183 Points
9 years ago
let the first term of an AP be a and the common difference be d
9th term=a+8d
13th term=a+12d
15th term=a+14d
for a gp the first term -a+8d
2 term-a+12d 3rd term-a+14d
we have (a+12d)^2=(a+8d)(a+14d)
a=-16d ----1
sum of infinite gp =1st term/1-r
=a+8d/(1-a+12d/a+8d) =80
putting the value of a=-16d in the anove equation we get d=4.75 and a=-76
common difference=4.75,common ratio =2
first term =-76
first term of gp=-38

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