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Q.
The sum of the products of the corresponding terms of the sequences $2,4,8,16,32$ and $128,32,8$, $2, \frac{1}{2}$ is
Sequences and Series
Solution:
Given, sequences are $2, 4, 8, 16, 32 ....$(i)
and $128,32,8,2, \frac{1}{2} ....$(ii)
Multiplying the corresponding terms of (i) and (ii) to obtain a new sequence $256,128,64,32,16$.
Let $ S=256+128+64+32+16$
Here, $ a=256, r=\frac{1}{2}$
$ \therefore $ Required sum $ S =\frac{256\left[1-\left(\frac{1}{2}\right)^5\right]}{1-\frac{1}{2}}$
$\left[\because S_n=\frac{a\left(1-r^n\right)}{1-r}\right.$ as $\left.r \propto 1\right]$
$=256 \times 2\left(1-\frac{1}{2^5}\right)$
$ =512 \times\left(1-\frac{1}{32}\right) $
$ =512\left(\frac{32-1}{32}\right) $
$ =16 \times 31 $
$ =496$