Question

If \( \frac{1}{2 \cdot 3^{10}}+\frac{2}{2^{2} \cdot 3^{9}}+\frac{3}{2^{3} \cdot 3^{8}}+\ldots++\frac{10}{2^{10} \cdot 3}=\frac{1}{2^{a}}+\frac{2}{3^{b}} \), then \( a+b \) is 10 15 17 22

Answer 1

\(S = \frac 1{2.3^{10}} + \frac 2{2^2.3^9} + \frac 3{2^3.3^8} + ...+ \frac{10}{2^{10}.3}\)

\(= \frac 1{2.3^{10}} \left(1 + 2(\frac 32) + 3(\frac 32)^2 + ....+10(\frac 32)^9\right)\)

\(= \frac 1{2.3^{10}} \left(\frac{11.(\frac 32)^{10} (\frac 32 -1) -(\frac 32)^{11} + 1}{(\frac 32 - 1)^2}\right)\)

\(= \frac {2^2}{2.3^{10}} \left(\frac {11}2 .(\frac 32)^{10} - (\frac 32)^{10} - \frac 12\right)\)

\(= \frac{11}{2^{10}} - \frac 2{2^{10}} - \frac 1{3^{10}}\)

\( = \frac 9{2^{10}} - \frac 1{3^{10}}\)

Not found in required form \(\frac 1{2a} + \frac 2{3b}\).