Saturday, 29 October 2011

Sequence-1 (CAT-2002)

Let S = 2x+5x2+9x3+14x4+20x5+…    infinity. The coefficient of nth 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
For Answer Click on "Read more" below:
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)

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