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Question

If S=49x+16x225x3+36x449x5+....... Find the value of S for x=12.

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Solution

This series is not AP,GP,HP and AGP
But we see 4,9,16,25....... are square numbers and 1,(x),(x)2,(x)3,(x)4,....... is a geometric progression. if we subtract square numbers. We can form an AP.
So, multiply given series by common ratio and then subtract it with original series.
Let s=49x+16x225x3+36x449x5+...... -----(1)
Multiply by common difference in equation 1
xs=4x+9x216x3+25x436x5+...... -----(2)
) On subtracting equation 2 from equation 1
We get,
s(1+x)=45x+7x29x3+11x413x5+...... -----(3)
Multiply by common difference in equation 3
s(1+x)x=4x+5x27x3+9x411x5+...... -----(4)
On subtracting equation 4 from equation 3
s(1+x)2=4x+2x22x3+2x42x5+......
s(1+x)2=4x+2(x2x3+x4x5+......)
s(1+x)2=4x+2x2(1x+x2......)
s(1+x)2=4x+2x2.11+x
s(1+x)2=4x+2x21+x=4+3x+x2x+1
s=x2+3x+4(x+1)3
When x=-12
s=(12)2+3(12)+4(12+1)3
1432+413=22


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