1 - 22 - 322 - 423 - - to n terms-

S=1+ 2 2 + 3 2 ^ 2 + 4 2 ^ 3 +.... n 2 ^ n 1 S 2 = 1 2 + 2 2 ^ 2 + 3 2 ^ 3 +.... n 1 2 ^ n 1 + n 2 ^ n S S 2 =1+ 1 2 + 1 ...

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Byju's Answer

Standard XII

Mathematics

Binomial Expression

1 + 22 + 322 ...

Question

$1 + \frac{2}{2} + \frac{3}{2^{2}} + \frac{4}{2^{3}} + . . . . . . . .$ to n terms.

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Solution

$S = 1 + \frac{2}{2} + \frac{3}{2^{2}} + \frac{4}{2^{3}} + . . . . \frac{n}{2^{n - 1}} \frac{S}{2} = \frac{1}{2} + \frac{2}{2^{2}} + \frac{3}{2^{3}} + . . . . \frac{n - 1}{2^{n - 1}} + \frac{n}{2^{n}} S - \frac{S}{2} = 1 + \frac{1}{2} + \frac{1}{2^{2}} + \frac{1}{2^{3}} + . . . . + \frac{1}{2^{n - 1}} - \frac{n}{2^{n}} \frac{S}{2} = \frac{(1 - \frac{1}{2^{n}})}{(1 - \frac{1}{2})} - \frac{n}{2^{n}} \frac{S}{2} = \frac{2 (2^{n} - 1)}{2^{n}} - \frac{n}{2^{n}} S = \frac{4 (2^{n} - 1) - 2 n}{2^{n}} = \frac{2^{n + 1} - n - 2}{2^{n - 1}}$

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1+22+322+423+........ to n terms.

S=1+22+322+423+....n2n−1S2=12+222+323+....n−12n−1+n2nS−S2=1+12+122+123+....+12n−1−n2nS2=(1−12n)(1−12)−n2nS2=2(2n−1)2n−n2nS=4(2n−1)−2n2n=2n+1−n−22n−1

$1 + \frac{2}{2} + \frac{3}{2^{2}} + \frac{4}{2^{3}} + . . . . . . . .$ to n terms.