Question 19

Find the sum of the products of the corresponding terms of the sequences 2, 4, 8, 16, 32 and 128, 32, 8, 2, $1 half$ .

Answer

Required sum = 2 × 128 + 4 × 32 + 8 × 8 + 16 × 2 + 32 × $1 half$

$equals space 64 space open square brackets 4 space plus space 2 space plus space 1 space plus 1 half space plus space 1 over 2 squared close square brackets$

Here, 4, 2, 1, $1 half comma space 1 over 2 squared$ is a G.P.

First term, a = 4

Common ratio, r = $1 half$

It is known that,

$S subscript n space equals space fraction numerator a space open parentheses 1 minus r to the power of n close parentheses over denominator 1 minus r end fraction therefore space S subscript 5 space equals space fraction numerator 4 open square brackets 1 minus open parentheses begin display style 1 half end style close parentheses to the power of 5 close square brackets over denominator 1 minus begin display style 1 half end style end fraction equals fraction numerator 4 open square brackets 1 minus begin display style 1 over 32 end style close square brackets over denominator begin display style 1 half end style end fraction equals fraction numerator 8 open parentheses 32 minus 1 close parentheses over denominator 32 end fraction equals 31 over 4 therefore space R e q u i r e d space s u m space equals space 64 space open parentheses 31 over 4 close parentheses space equals space open parentheses 16 close parentheses open parentheses 31 close parentheses space equals 496$