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Class 11
Mathematics
Sequence and Series
let s be the sum p the product and r the sum of r...

Question:
Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P²Rⁿ = Sⁿ

Answer:

Let the G.P. be a, ar, ar², ar³, … ar^{n – 1}…

According to the given information,

$S space equals space fraction numerator a space open parentheses r to the power of n space minus 1 close parentheses over denominator r space minus 1 end fraction P space equals space a to the power of n space end exponent cross times space r to the power of 1 plus 2 plus... plus n minus 1 end exponent space space space space equals space a to the power of n space end exponent r to the power of fraction numerator n open parentheses n minus 1 close parentheses over denominator 2 end fraction space space space space space space space open square brackets because space S u m space o f space f i r s t space n space n a t u r a l space n u m b e r s space i s space fraction numerator n open parentheses n plus 1 close parentheses over denominator 2 end fraction close square brackets end exponent R space equals space 1 over a space plus space fraction numerator 1 over denominator a r end fraction space plus space... space plus space fraction numerator 1 over denominator a r to the power of n minus 1 end exponent end fraction space space space space equals space fraction numerator r to the power of n minus 1 end exponent plus r to the power of n minus 2 end exponent space plus space... r plus 1 over denominator a r to the power of n minus 1 end exponent end fraction space space space space equals space fraction numerator 1 open parentheses r to the power of n space minus 1 close parentheses over denominator open parentheses r minus 1 close parentheses end fraction space cross times space fraction numerator 1 over denominator a r to the power of n minus 1 end exponent end fraction space space space space space space space space space space open square brackets because space 1 comma r comma... r to the power of n minus 1 end exponent space f o r m s space a space G. P. close square brackets space space space space equals space fraction numerator r to the power of n space minus 1 over denominator a r to the power of n minus 1 end exponent open parentheses r minus 1 close parentheses end fraction therefore space P squared R to the power of n space equals space a to the power of 2 n end exponent r to the power of n open parentheses n minus 1 close parentheses space end exponent fraction numerator open parentheses r to the power of n space minus 1 close parentheses to the power of n over denominator a to the power of n r to the power of n open parentheses n minus 1 close parentheses end exponent open parentheses r minus 1 close parentheses to the power of n end fraction space space space space space space space space space space space space space space space space space equals space fraction numerator a to the power of n open parentheses r to the power of n minus 1 close parentheses to the power of n over denominator open parentheses r minus 1 close parentheses to the power of n end fraction space space space space space space space space space space space space space space space space space equals space open square brackets fraction numerator a open parentheses r to the power of n space minus 1 close parentheses over denominator open parentheses r minus 1 close parentheses end fraction close square brackets to the power of n space space space space space space space space space space space space space space space space equals space S to the power of n$

Hence P² Rⁿ= S_n

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Question:
Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P²Rⁿ = Sⁿ

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Question: Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P2Rn = Sn

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Question:
Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P²Rⁿ = Sⁿ