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Q) A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying odd places then find the common ratio.
Asked by Anish | 14 Apr, 2019, 15:53: PM
answered-by-expert Expert Answer
I t space i s space g i v e n space t h a t space t h e space n u m b e r space o f space t e r m s space i s space e v e n S o comma space l e t space t h e space G. P space h a s space 2 n space t e r m s T h e r e f o r e space t h e space G. P. space i s space g i v e n space b y a comma space a r comma space a r squared comma space...... comma space a r to the power of 2 n minus 1 end exponent space............ space left parenthesis i right parenthesis N o w comma space t h e space t e r m s space o c c u p y i n g space o d d space p l a c e s space a r e space a comma space a r squared comma space a r to the power of 4 comma space..... comma space a to the power of 2 n minus 2 end exponent space... left parenthesis i i right parenthesis space w h i c h space a l s o space f o r m s a space G. P. space w i t h space f i r s t space t e r m space equals space a space a n d space c o m m o n space r a t i o equals r squared S i n c e comma space i t space i s space g i v e n space t h a t space t h e space s u m space o f space a l l space t h e space t e r m s space o f space left parenthesis i right parenthesis space i s space 5 space t i m e s space t h e space s u m space o f space a l l space t h e space t e r m s space o f space left parenthesis i i right parenthesis A n d space a l s o space w e space k n o w space t h a t space t h e space s u m space o f space n space t e r m s space o f space a space G. P. space i s space a times fraction numerator r to the power of n minus 1 over denominator r minus 1 end fraction rightwards double arrow a times fraction numerator r to the power of 2 n end exponent minus 1 over denominator r minus 1 end fraction equals 5 cross times a times fraction numerator left parenthesis r squared right parenthesis to the power of n minus 1 over denominator r squared minus 1 end fraction equals 5 a times fraction numerator r to the power of 2 n end exponent minus 1 over denominator r squared minus 1 end fraction rightwards double arrow a times fraction numerator r to the power of 2 n end exponent minus 1 over denominator r minus 1 end fraction equals 5 a times fraction numerator r to the power of 2 n end exponent minus 1 over denominator r squared minus 1 end fraction rightwards double arrow fraction numerator 1 over denominator r minus 1 end fraction equals fraction numerator 5 over denominator left parenthesis r plus 1 right parenthesis left parenthesis r minus 1 right parenthesis end fraction rightwards double arrow r plus 1 equals 5 rightwards double arrow r equals 4
Answered by Renu Varma | 15 Apr, 2019, 12:41: PM

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