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Question
Find the sum of the following series 1+32n+12n−1+5(2n+12n−1)2+.... to n terms.
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Solution
⇒Sn=1+3x+5x2+...+(2n−1)xn−1−2xSn=−2x−6x2−....−(4n−6)xn−1−(4n−2)xnx2Sn=x2+x3+....+(2n−5)xn−1+(2n−3)xn+(2n−1)xn+1(1−x)2Sn=1+x−(2n+1)xn+(2n−1)xn+1The sum of the series follow when x=2n+12n−1(−22n−1)2S=1+2n+12n−1−(2n+12n−1)n{(2n+1)−(2n+1)}4(2n−1)2S=4n(2n−1)S=n(2n−1)
⇒Sn=1+3x+5x2+...+(2n−1)xn−1
−2xSn=−2x−6x2−....−(4n−6)xn−1−(4n−2)xn
x2Sn=x2+x3+....+(2n−5)xn−1+(2n−3)xn+(2n−1)xn+1
(1−x)2Sn=1+x−(2n+1)xn+(2n−1)xn+1
The sum of the series follow when x=2n+12n−1
(−22n−1)2S=1+2n+12n−1−(2n+12n−1)n{(2n+1)−(2n+1)}
4(2n−1)2S=4n(2n−1)
S=n(2n−1)
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