Sequence-1 (CAT-2002)

Let S = 2x+5x²+9x³+14x⁴+20x⁵+…    infinity. The coefficient of n^th term is n(n+3)/2. The sum S is:

(1)x(2-x)/(1-x)³  (2) (2-x)/(1-x)³   (3) x(2-x)/(1-x)²   (4)None of these

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Solution:

It is a standard model which deals with manipulation and simplification of equations.

S = 2x+5x²+9x³+14x⁴+20x⁵+….       -----(1)
Multiplying on both sides with 'x':

xS = 2x²+5x³+9x⁴+14x⁵+20x⁶+….   -----(2)

(1)-(2) gives

(1-x)S = 2x+3x²+4x³+5x⁴+…           ------(3)

Multiplying on both sides with 'x':
x(1-x)S = 2x²+3x³+4x⁴+5x⁵+…         ------(4)

(3)-(4) gives

(1-x)(1-x)S = 2x+x²+x³+x⁴+…

(1-x)²S = x+(x+x²+x³+x⁴+…... infinite terms) 

(1-x)²S = x + (sum of G.P with initial term ‘x’ and common ratio ‘x’)                 

“Formula: sum of infinite G.P = a/(1-r), where a is initial term and r is common ratio”

(1-x)²S = x + x/(1-x) = (x-x²+x)/(1-x) = (2x-x²)/(1-x) = x(2-x)/(1-x)

=> S = x(2-x)/(1-x)³

Answer (1)