1 1!(n - 1) ,! + 1 3!(n - 3)! + 1 5!(n - 5)! + .... = - Vidyadip

MCQ

$\frac{1}{{1!(n - 1)\,!}} + \frac{1}{{3!(n - 3)!}} + \frac{1}{{5!(n - 5)!}} + .... = $

A
$\frac{{{2^n}}}{{n!}}$; for all even values of $n$
✓
$\frac{{{2^{n - 1}}}}{{n!}}$; for all values of $n$ i.e., all even odd values
C
$0$
D
None of these

✓

Answer

Correct option: B.

$\frac{{{2^{n - 1}}}}{{n!}}$; for all values of $n$ i.e., all even odd values

b
(b) Multiplying each term by $n!$ the question reduces to

$\frac{{n!}}{{1!(n - 1)!}} + \frac{1}{{3!}}.\frac{{n!}}{{(n - 3)\,!}} + \frac{1}{{5!}}.\frac{{n!}}{{(n - 5)!}} + ....$

$ = {\,^n}{C_1} + {\,^n}{C_3} + {\,^n}{C_5} + .... = {2^{n - 1}}$.

Thus $\frac{1}{{1!(n - 1)!}} + \frac{1}{{3!(n - 3)!}} + \frac{1}{{5!(n - 5)!}} + ....$$ = \frac{1}{{n!}}{2^{n - 1}}$.

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