Prove that if n be an even integer- 1- 1 -n-1 - 1-3 -n-3 - 1-5 -n-5 - - - 1-n-1 -1 - 2n-1 -n-

We know that #160; (1+x)^n=^nC_0+ ^nC_1x+ ^nC_2x^2+ ^nC_3x^3 . . . . .+ ^nC_n.x^n (1 x)^n=^nC_0 ^nC_1x+ ^nC_2x^2 ^nC_3x^3 . . . . .+ ( 1)^n ^nC_n x ...

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Standard XII

Mathematics

Special Integrals - 1

Prove that if...

Question

Prove that if $n$ be an even integer,
$\frac{1}{| 1 - | n - 1 - ---- -} + \frac{1}{| 3 - | n - 3 - ---- -} + \frac{1}{| 5 - | n - 5 - ---- -} + . . . . + \frac{1}{| n - 1 - ---- - | 1 -} = \frac{2^{n - 1}}{| n - -}$ .

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Solution

We know that
$(1 + x)^{n} =^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} +^{n} C_{3} x^{3} . . . . . +^{n} C_{n} . x^{n}$
$(1 - x)^{n} =^{n} C_{0} -^{n} C_{1} x +^{n} C_{2} x^{2} -^{n} C_{3} x^{3} . . . . . + (- 1)^{n}^{n} C_{n} x^{n}$

$\Rightarrow (1 + x)^{n} - (1 - x)^{n} = 2 [^{n} C_{1} . x +^{n} C_{3} . x^{3} . . . . . +^{n} C_{n - 1} . x^{n - 1}]$
Substitute $x = 1$ in both equations, we get
$2^{n} = 2 (^{n} C_{1} +^{n} C_{3} + . . . . . . . +^{n} C_{n - 1})$ ...... $(i)$
Consider $\frac{1}{n!} [\frac{n!}{1! (n - 1)!} + \frac{n!}{3! (n - 3)!} + \frac{n!}{5! (n - 5)!} + . . . . + \frac{n!}{(n - 1)! 1!}]$
$= \frac{1}{n!} [^{n} C_{1} +^{n} C_{3} +^{n} C_{5} + . . . . . +^{n} C_{n - 1}]$
$= \frac{1}{n!} [2^{n - 1}]$

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Prove that if n be an even integer, 1|1–|n−1––––––+1|3–|n−3––––––+1|5–|n−5––––––+....+1|n−1––––––|1–=2n−1|n––.

Prove that if $n$ be an even integer,
$\frac{1}{| 1 - | n - 1 - ---- -} + \frac{1}{| 3 - | n - 3 - ---- -} + \frac{1}{| 5 - | n - 5 - ---- -} + . . . . + \frac{1}{| n - 1 - ---- - | 1 -} = \frac{2^{n - 1}}{| n - -}$ .