Question

Find the sum of the products of the corresponding terms of the sequences $2,4,8,16,32$ and $128,32,8,2,$ $\frac{1}{2}$.

Answer 1

Required sum

$=2\times 128+4\times 32+8\times 8+16\times 2+32\times \frac{1}{2}=64[4+2+1+\frac{1}{2}+\frac{1}{2^2}]$

Here, $4,2,1,\frac{1}{2},\frac{1}{2^2}$ is a G.P. with First term ,$a=4$ and common ratio, $r=\frac{1}{2}$

It is known that, $S_n=\frac{a(1-r^n)}{1-r}$

$\therefore S_5=\frac{4[1+-{\left(\frac{1}{2}\right)}^5]}{1-\frac{1}{2}}=\frac{4[1-\frac{1}{32}]}{\frac{1}{2}}=8\frac{(32-1)}{32}=\frac{31}{4}$

Hence required sum $=64\left(\frac{31}{4}\right)=\left(16\right)\left(31\right)=496$