MCQ
$2 + 5 + 14 + 41 + ......$ શ્રેણીના $n$ પદોનો સરવાળો કેટલો થાય ?
- A$\frac{n}{2}\, + \,\frac{1}{4}\,({3^n}\, - \,1)$
- B$\frac{n}{2}\, + \,\frac{3}{4}\,({3^n}\, - \,1)$
- C$\frac{n}{2}\, + \,\frac{1}{2}\,({3^n}\, - \,1)$
- Dઆપેલ પૈકી એકપણ નહિ.
ક્રમિક પદોનો તફાવત $3, 9, 27, ...$
$t_n= 2 + 3 + 9 + 27 .........n$પદો
$= 2 + [3 + 9 + 27 + ....... (n - 1)$ પદો]
$ = \,2\,\, + \frac{{3\,\left( {1\, - \,{3^{n - 1}}} \right)}}{{1\, - \,3}}$
$ = \,\,2\,\, - \,\,\frac{3}{2}\,\left( {1\, - \,{3^{n - 1}}} \right)\,\, = \,\,\frac{1}{2}\,\, + \,\,\frac{1}{2}\,.\,{3^n}$
$\therefore \,\,{S_n}\,\, = \,\,\Sigma {t_n}\, = \,\,\frac{1}{2}\Sigma 1\,\, + \,\,\frac{1}{2}\,\Sigma {3^n}$
$ = \,\,\frac{1}{2}.n\,\, + \,\,\frac{1}{2}\,\left( {3\, + \,{3^2}\, + \,{3^3}\, + \,...\, + \,{3^n}} \right)$
$ = \,\,\frac{n}{2}\,\, + \,\,\frac{1}{2}.\,\frac{{3\left( {1\, - \,{3^n}} \right)}}{{1 - 3}}\,\, = \,\,\frac{n}{2}\,\, + \,\,\frac{3}{4}\,\left( {{3^n} - 1} \right)$
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