Permutations And Combinations

Question
CBSEENMA11013075

The first term of GP. is 1. The sum of 3rd and 5th term is 90. Find the common ratio of G.P.

Solution

Let a be first term and r be the common ratio of G.P.
Given a = 1,  also,  straight t subscript 3 space plus space straight t subscript 5 space equals space 90
rightwards double arrow                  ar squared space plus space ar to the power of 4 space equals space 90
rightwards double arrow          space space straight r squared plus straight r to the power of 4 space equals space 90                                                                 (∵ a = 1)
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rightwards double arrow       space space straight r to the power of 4 plus 10 straight r squared minus 9 straight r squared minus 90 space equals space 0  or straight r squared left parenthesis straight r squared plus 10 right parenthesis space minus space 9 left parenthesis straight r squared plus 10 right parenthesis space equals space 0
or       left parenthesis straight r squared minus 9 right parenthesis space left parenthesis straight r squared plus 10 right parenthesis space equals space 0
rightwards double arrow            straight r squared space minus space 9 space space equals 0                                       left parenthesis straight r squared plus 10 equals 0 space space is space impossible right parenthesis
rightwards double arrow                  straight r space equals space plus-or-minus space 3

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