Permutations and Combinations

Question

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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Permutations And Combinations

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

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