Prove by induction the following result where N is a positive even integer. S_1 ^2 +S_2 ^2 =2^N where S_1 =Σ_(r=1) ^((N/2)+1) (−1)^(r−1) ((N),((2(r−1))) ) and S_2 =Σ_(r=1) ^(N/2) (−1)^(r+1) ((N),((2r−1)) ) .


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Question Number 1074 by Yugi last updated on 05/Jun/15

$P r o v e b y i n d u c t i o n t h e f o l l o w i n g r e s u l t w h e r e N i s a p o s i t i v e e v e n i n t e g e r .$ $S_{1}^{2} + S_{2}^{2} = 2^{N}$ $w h e r e S_{1} = \underset{r = 1}{\sum^{\frac{N}{2} + 1}} {(- 1)}^{r - 1} (\begin{matrix} N \\ 2 (r - 1) \end{matrix}) a n d S_{2} = \underset{r = 1}{\sum^{N / 2}} {(- 1)}^{r + 1} (\begin{matrix} N \\ 2 r - 1 \end{matrix}) .$
Commented by prakash jain last updated on 07/Jun/15

$^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$ $^{n + 2} C_{r} =^{n + 1} C_{r - 1} +^{n + 1} C_{r} =^{n} C_{r - 2} + 2^{n} C_{r - 1} +^{n} C_{r}$
Commented by prakash jain last updated on 07/Jun/15

$Use the above relation to write {\overset{N + 2}{S}}_{1} in terms$ $of {\overset{N}{S}}_{1} and {\overset{N}{S}}_{2} .$
Commented by 112358 last updated on 07/Jun/15

$T h a n k s .$
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