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find sum of 1+3x+6x^2+10x^3+15x^4 + ----...

find sum of `1+3x+6x^2+10x^3+15x^4 + -------oo` where ` |x| < 1 , x!= 0`

A

` 1/(1-x)^(2)`

B

` 1/(1-x)`

C

` 1/(1+x)^(2)`

D

` 1/(1-x)^(3)`

Text Solution

Verified by Experts

The correct Answer is:
d
Let = `1+ 3x + 6x^(2) +10x^(3) +…" "` … (i)
` rArr S(1-x)=1 + 2 x + 3x^(2) + 4x^(3)+…." "` .. .(ii)
On subtracting Eq. (ii) from Eq. (i), we get
` S (1-x) = 1+2x +3x^(2) +4x^(2) + … oo )" "` …(iii)
` rArr x(1-x)5 = x + 2x^(2) + 3x^(3) + ...oo " "` ...(iii)
` rArr x (1-x) 5 = x +2x^(2)+3x^(3) + ... oo" "` ...(iv)
Again subtracting Eq. (iv) from Eq. (iii) , we get ,
` S [(1-x)-x(1-x)] = (1+x+x^(2)+x^(3)+...oo)`
` rArr S [ (1-x ) (1-x)] = 1/(1-x)`
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