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If S be the sum, P be the product and R the sum of the reciprocal , of n terms in a G.P. Prove that P 2 =(S/R) n

If S be the sum, P be the product and R the sum of the reciprocal , of n terms in a G.P. Prove that    P2=(S/R)n

Grade:11

1 Answers

Arun
25763 Points
4 years ago
Let a be the first term and R the common ratio. 

Then, we have 

s = a + aR + aR² + ... to n terms = a ( 1 - Rⁿ ) / ( 1 - R ) .................... (1) 

r = (1/ a ) + (1/ aR ) + ... 

= (1/a) [ (1/R)ⁿ - 1 ] / [ (1/R) - 1 ] 

= ( R/ aRⁿ ) · ( 1 - Rⁿ ) / (1 - R ) 

= (R/ aRⁿ) · ( s / a ) = r ...................... from (1) 

∴ s / r = a² Rⁿ / R = a² Rⁿֿ¹ 

∴ ( s / r )ⁿ = a²ⁿ · Rⁿ⁽ⁿֿ¹⁾ ............................. (2) 
___________________________ 

P = 

= (a) (aR) (aR²) ... (aRⁿֿ¹) 

= aⁿ · R¹ ⁺ ² ⁺ ³ ⁺ ... ⁺ ⁽ⁿֿ¹⁾ 

= aⁿ · Rⁿ⁽ⁿֿ¹⁾ʹ² 

Hence, 

P² = a²ⁿ · Rⁿ⁽ⁿֿ¹⁾ ........................................... (3) 
_________________________ 

From (2) and (3), 

P² = ( s/r )ⁿ 
Hence proved

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