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Question Number 13237 by Abbas-Nahi last updated on 17/May/17

Commented by RasheedSindhi last updated on 17/May/17

$\frac{36}{(n + 3)!} - \frac{1}{(n + 1)!} - \frac{1}{n!} = 0$ $\frac{1}{n!} (\frac{36}{(n + 3) (n + 2) (n + 1)} - \frac{1}{(n + 1)} - 1) = 0$ $\frac{1}{n!} = 0 ∣ (\frac{36}{(n + 3) (n + 2) (n + 1)} - \frac{1}{(n + 1)} - 1) = 0$ $\frac{1}{n!} = 0 \Rightarrow n \to \infty$
Commented by Abbas-Nahi last updated on 17/May/17

$s o r r y t h e a n s w e r i s w r o n g! n ⇏ \infty$
	Answered by RasheedSindhi last updated on 17/May/17

	$\frac{36}{(n + 3)!} = \frac{1}{(n + 1)!} + \frac{1}{n!}$ $\frac{36}{(n + 3) (n + 2) (n + 1) (n!)} = \frac{1}{(n + 1) (n!)} + \frac{1}{n!}$ $\frac{36}{(n + 3) (n + 2) (n + 1)} = \frac{1}{(n + 1)} + \frac{1}{1}$ $\frac{36}{(n + 3) (n + 2) (n + 1)} = \frac{1 + (n + 1)}{(n + 1)}$ $\frac{36}{(n + 3) (n + 2)} = n + 2$ $(n + 3) {(n + 2)}^{2} = 36$ $See {mrW 1}^{'} s comment below .$
	Commented by mrW1 last updated on 17/May/17

	$36 = (n + 2) (n + 2) (n + 3)$ $f o r n = i n t e g e r a n d$ $36 = 3 \times 3 \times 4$ $I t h i n k t h e o n l y s o l u t i o n i s n = 1 .$ $U p o n n ⩾ 2$ $(n + 2) (n + 2) (n + 3) ⩾ 4 \times 4 \times 5 = 80 > 36$
	Commented by Abbas-Nahi last updated on 17/May/17

	$N ICE t r y s i r!$ $b u t f i n a l s t e p i s$ $(n + 3) {(n + 2)}^{2} = 36$ $(n + 3) {(n + 2)}^{2} = 4 \times 9; & b y c o m p a r i n g$ $(n + 3) = 4 \Rightarrow n = 1$ $\lor {(n + 2)}^{2} = 9 \Rightarrow n + 2 = \bar{+} 3$ $n = 1$ $\lor$ $n = - 5 \times$ $c h e c k i t n = 1$ $\frac{36}{(1 + 3)!} = \frac{1}{(1 + 1)!} + \frac{1}{1!}$
	Commented by Abbas-Nahi last updated on 17/May/17

	$t h a n k y o u s i r!$
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