$\begin{array}{1 1} 496 \\ 248 \\ 992 \\ 1984 \end{array} $

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Given sequences are

$2,4,8,16,32$.........(i) and

$128,32,8,2,\large\frac{1}{2}$............(ii)

Both the sequences have 5 terms and are G.P.

We have to find the sum of the products of the corresponding terms

$i.e.,$

$(2\times 128)+(4\times 32)+(8\times 8)+(16\times 2)+(32\times \large\frac{1}{2})$

This can be written as

$=2^8+2^7+2^6+......2^4$

It is again a G.P. with first trerm $a=2^8$ and

common ratio $r=\large\frac{2^7}{2^8}=\frac{1}{2}$

The no. of terms of this G.P=5

$\therefore\:$ The required sum $=S_5=a\large\frac{1-r^5}{1-r}$

$=2^8.\large\frac{1-1/2^5}{1-1/2}=$$2^8.\large\frac{\large\frac{2^5-1}{2^5}}{\large\frac{1}{2}}$

$=2^4(2^5-1)=16\times 31=496$

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