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શ્નેણી $\frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{{15}}{{16}} + .........$ $n$ પદનો સરવાળો મેળવો.
${2^n} - n - 1$
$1 - {2^{ - n}}$
$n + {2^{ - n}} - 1$
${2^n} - 1$
Solution
(c) The sum of the first $n$ terms is
${S_n} = \left( {1 – \frac{1}{2}} \right) + \left( {1 – \frac{1}{{{2^2}}}} \right) + \left( {1 – \frac{1}{{{2^3}}}} \right) + \left( {1 – \frac{1}{{{2^4}}}} \right)+$$ …… + \left( {1 – \frac{1}{{{2^n}}}} \right)$
$ = n – \left\{ {\frac{1}{2} + \frac{1}{{{2^2}}} + ….. + \frac{1}{{{2^n}}}} \right\}$
= $n – \frac{1}{2}\left( {\frac{{1 – \frac{1}{{{2^n}}}}}{{1 – \frac{1}{2}}}} \right) = n – \left( {1 – \frac{1}{{{2^n}}}} \right) = n – 1 + {2^{ – n}}$.
Trick : Check for $n = 1,\;2\;$
$i.e.$ ${S_1} = \frac{1}{2},\;{S_2} = \frac{5}{4}$
and $(c)$ $ \Rightarrow {S_1} = \frac{1}{2}$
and ${S_2} = 2 + {2^{ – 2}} – 1 = \frac{5}{4}$.
