If s be the sum p be the product

Asked by admin @ 13/08/2021 in Math viewed by 281 People

Question 14 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

Class X1 - Maths -Sequences and Series Page 199

Answered by admin @ 13/08/2021

Let the GP is a , ar , ar² , ar³ , .......... arⁿ⁻¹.
S = Sum of n terms
= a + ar + ar² + ar³ + .......+ arⁿ⁻¹
= a( rⁿ - 1)/( r - 1)

and R = sum of reciprocal of n terms
= 1/a + 1/ar + 1/ar² + 1/ar³ + ......... + 1/arⁿ⁻¹
= 1/a[ 1 - ( 1/r )ⁿ ]/( 1 - 1/r )
= ( rⁿ - 1)r/arⁿ(r - 1)

P = product of n terms = a × ar × ar² × ar³ × .....× arⁿ⁻¹
=〈 a¹⁺¹⁺¹⁺¹⁺¹⁺¹⁺¹⁺⁻⁻¹〉〈r¹⁺²⁺³⁺⁻⁻⁻⁻⁽ⁿ⁻¹⁾〉
= aⁿ. r^{n(n-1)/2}
take square both sides,
P² = a²ⁿrⁿ⁽ⁿ⁻¹⁾

Now, we have to prove P²Rⁿ = Sⁿ
or , P² = (S/R)ⁿ

RHS = (S/R)ⁿ
= [a(rⁿ - 1)/(r - 1) ]/[(rⁿ - 1)r/arⁿ(r - 1) ]ⁿ
= [a(rⁿ - 1)/(r - 1) × arⁿ (r - 1)/(rⁿ - 1)r ]ⁿ
=〈 a²rⁿ⁻¹ 〉ⁿ
= a²ⁿ rⁿ⁽ⁿ⁻¹⁾ = P² = LHS

If s be the sum p be the product

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