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  4. The sum of the series 1 + 2.2 + 3.22 + 4.23+ 5.24 + .......... + 100.299 is

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Q. The sum of the series $1 + 2.2 + 3.2^2 + 4.2^3+ 5.2^4 + .......... + 100.2^{99}$ is

Sequences and Series

Solution:

Let $S = 1+ 2.2+3.2^{2} +4.2^{3} + ...... + 100.2^{99} $
$ \therefore 2S = 1.2+2.2^{2} +3.2^{3}+......+99.2^{99}+100.2^{100} $
Subtracting, we get
$ -S = 1+1.2+1.2^{2}+1.2^{3}+..... +1.2^{99 }-100.2^{100} $
$= \left(1+2+2^{2}+......2^{99} \right)-100.2^{100}$
$ = \frac{1\left(2^{100}-1\right)}{2-1} -100.2^{100} = 2^{100} -1 -100.2^{100} $
$= \frac{1\left(2^{100}-1\right)}{2-1} -100.2^{100}$
$ = 2^{100} -1 -100.2^{100}$
$\therefore S = 100.2^{100} -2^{100}+1$
$ = 99.2^{100} +1$