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Question Number 57363 by Tawa1 last updated on 03/Apr/19

Commented by Tawa1 last updated on 03/Apr/19

$Please help . my problem is from \frac{2^{m} . m (m + 1) (m + 2) . . . . (2 m - 1)}{1.3.5 . . . . . . . . (2 m - 1)}$ $now why did they multiply Numerator and denominator by$ $(m - 1)! . 2 m to get . . .$ $\frac{2^{m} . m (m + 1) (m + 2) . . . . (2 m - 1) \times (m - 1)! . 2 m}{[1.3.5 . . . . . . . . (2 m - 1)] . . . . . . . . (m - 1)! . 2 m}$ $And suddenly becomes$ $\frac{2^{m} . (2 m)!}{[1.3.5 . . . . . . . . (2 m - 1)] . . . . . . . . (m - 1)! . 2 m}$ $And later .$ $\frac{2^{m} . 2^{m} m! [1.3.5 . . . . . (2 m - 1)]}{[1.3.5 . . . . . . . . (2 m - 1)] . . . . . . . . (m - 1)! . 2 m}$ $So, i {don}^{'} t understand all the steps jump .$ $I {don}^{'} t want to cram, i want to understand the steps because$ $of exam .$
Commented by tanmay.chaudhury50@gmail.com last updated on 03/Apr/19

$l o o k a t t h e D_{r} = 1.3.5 . . . . (n - 1)$ $D_{r} = (n - 1) . . . 5.3.1 \leftarrow l o o k e v e n n u m b e r a r e m i s s i n g$ $s o D_{r} = [(n - 1) (n - 2) (n - 3) (n - 4) . . 6.5.4.3.2.1] \times \frac{1}{(n - 2) (n - 4) . . 6.4.2}$ $h e r e i h a v e i n s e r t e d e v e n n u m b e r s a n d d e v i d e d$ $b y s a m e e v e n n u m b e r s t o k e e p t h e v a l u e o f$ $D_{r} = f i x e d$ $D_{r} = \frac{1}{(n - 2) (n - 4) . . . 6.4.2} \times (n - 1) (n - 2) (n - 3) . . 6.5.4.3.2.1$ $D_{r} = \frac{(n - 1)!}{(n - 2) (n - 4) . . 6.4.2}$ $n o w$ $\frac{N_{r}}{D_{r}} = \frac{n (n + 2) (n + 4) . . (2 n - 2)}{\frac{(n - 1)!}{(n - 2) (n - 4) . . 6.4.2}}$ $\frac{N_{r}}{D_{r}} = \frac{(2 n - 2) (2 n - 4) . . . (n + 4) (n + 2) n \times (n - 2) (n - 4) (n - 6) . . 6.4.2}{(n - 1) \cdot}$ $l e t n = 2 k$ $\frac{N_{r}}{D_{r}} = \frac{(4 k - 2) (4 k - 4) . . (2 k + 4) (2 k + 2) (2 k) (2 k - 2) (2 k - 4) (2 k - 6) . . 6.4.2}{(2 k - 1)!}$ $n o w i n A P s e r i e s f i r s t t e r m = 2$ $c o m m o n d i f f e r e n c e = + 2$ $T_{r} = 2 + (r - 1) 2 \to 2 r$ $2 r = 4 k - 2 h e n c e r = 2 k - 1$ $\frac{N_{r}}{D_{r}} = \frac{2 (2 k - 1) \times 2 (2 k - 2) \times . . 2 (k + 2) \times 2 (k + 1) \times 2 (k) \times 2 (k - 1) \times 2 (k - 3) . . 2 \times (3) \times 2 (2) \times 2 (1)}{(2 k - 1)!}$ $n o w h o w m a n y 2 a r e m u l t i p l i e d i n N_{r} . .$ $w e h a v e c a c u l a t e d u s i n g A P s e r i e s f o r m u l a$ $t h a t i s r \to [r = 2 k - 1]$ $s o \frac{N_{r}}{D_{r}} = \frac{2^{2 k - 1} \times (2 k - 1) (2 k - 2) \times . . (k + 2) (k + 1) (k) (k - 1) . . 3.2.1}{(2 k - 1)}$ $= \frac{2^{2 k - 1} \times (2 k - 1)!}{(2 k - 1)!}$ $= 2^{2 k - 1}$ $= 2^{n - 1} [n = 2 k w e c l n s i d e r e d]$
Commented by Tawa1 last updated on 03/Apr/19

$Wow . God bless you sir .$
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