Question 12

The sum of first three terms of a G.P. is 39 over 10 and their product is 1. Find the common ratio and the terms.

Answer

Let a over r space comma space a comma space a r be the first three terms of the G.P.

a over r space plus space a space plus space a r space equals space 39 over 10 space space space space space space... space left parenthesis 1 right parenthesis
open parentheses a over r close parentheses open parentheses a close parentheses open parentheses a r close parentheses space equals space 1 space space space space space space space space space space space space space space... space left parenthesis 2 right parenthesis

From (2), we obtain

a3 = 1

⇒ a = 1 (Considering real roots only)

Substituting a = 1 in equation (1), we obtain

1 over r space plus space 1 space plus space r space equals space 39 over 10
rightwards double arrow 1 space plus space r space plus space r squared space equals space 39 over 10 r
rightwards double arrow 10 space plus space 10 r space plus space 10 r squared space minus 39 r space equals 0
rightwards double arrow 10 r squared space minus 29 r space plus space 10 space equals 0
rightwards double arrow space 10 r squared space minus space 25 r space minus space 4 r space plus space 10 space equals space 0
rightwards double arrow 5 r space open parentheses 2 r space minus 5 close parentheses space minus space 2 open parentheses 2 r space minus space 5 close parentheses space equals space 0
rightwards double arrow open parentheses 5 r space minus space 2 close parentheses open parentheses 2 r minus 5 close parentheses space equals space 0
rightwards double arrow r space equals space 2 over 5 space o r space 5 over 2
T h u s comma space t h e space t h r e e space t e r m s space o f space G. P. space a r e space 5 over 2 comma space 1 space comma space a n d space 2 over 5. space space

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