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Sum of n Terms

In a G.P., th...

Question

In a G.P., the ratio between the sum of first three terms and that of the first six terms is 125 : 152.

Find its common ratio.

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Solution

$L e t space apostrophe a apostrophe space b e space f i r s t space t e r m space a n d space apostrophe r apostrophe space c o m m o n space r a t i o space o f space G. P S â�� space divided by space S â�� space equals space 125 over 152 space fraction numerator a space left parenthesis space r Â³ space minus space 1 space right parenthesis over denominator space left parenthesis space r space minus space 1 space right parenthesis end fraction space space divided by space fraction numerator a space left parenthesis space r space â�¶ space minus space 1 right parenthesis over denominator left parenthesis space r minus space 1 right parenthesis end fraction equals space fraction numerator 125 over denominator space 152 space end fraction fraction numerator left parenthesis space r Â³ space minus space 1 space right parenthesis over denominator left parenthesis space r space â�¶ space minus space 1 space right parenthesis space end fraction space space equals space fraction numerator 125 over denominator space 152 end fraction space fraction numerator left square bracket space r Â³ space minus space 1 space right square bracket over denominator left square bracket space left parenthesis r Â³ right parenthesis space Â² space minus space left parenthesis 1 right parenthesis Â² space right square bracket space end fraction space equals fraction numerator space 125 over denominator space 152 end fraction space fraction numerator left square bracket space r Â³ space minus space 1 space right square bracket over denominator space left square bracket space left parenthesis space r Â³ space minus space 1 right parenthesis space left parenthesis r Â³ space plus space 1 space right parenthesis space end fraction space equals space fraction numerator 125 over denominator 152 space end fraction space fraction numerator 1 over denominator space left square bracket space r Â³ space plus space 1 space right square bracket end fraction space space equals space fraction numerator 125 over denominator 152 space space end fraction space 152 space equals space 125 space left square bracket space r Â³ space plus space 1 space right square bracket space 152 space equals 125 space r Â³ space plus space 125 space 27 over 125 space space space equals space r space Â³ space space open parentheses 3 over 5 close parentheses Â³ space equals space r space Â³ space r equals space 3 over 5$
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