Without using arithmatic Series concept or formula prove the following: 1+2+3+...+n=((n(n+1))/2)


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Question Number 2759 by RasheedAhmad last updated on 26/Nov/15

$W i t h o u t u s i n g a r i t h m a t i c S e r i e s$ $c o n c e p t o r f o r m u l a p r o v e t h e f o l l o w i n g :$ $1 + 2 + 3 + . . . + n = \frac{n (n + 1)}{2}$
	Answered by 123456 last updated on 26/Nov/15

	$induction$ $1 + 2 + \cdot \cdot \cdot + n = \frac{n (n + 1)}{2}$ $base case : n = 1$ $\frac{1 (1 + 1)}{2} = \frac{2}{2} = 1$ $induction step :$ $suppose its true for n, lets shown its true$ $for n + 1$ $1 + 2 + \cdot \cdot \cdot + n + n + 1 = \frac{n (n + 1)}{2} + n + 1$ $= (n + 1) (\frac{n}{2} + 1)$ $= \frac{(n + 1) (n + 2)}{2}$ $= \frac{(n + 1) [(n + 1) + 1]}{2}$ $so, since its true for n = 1, then its true$ $to n = 1, 2, 3, 4, . . . .$ $n \in N^{*}$
	Commented by Rasheed Soomro last updated on 26/Nov/15

	$N i c e!$ $A c t u a l l y a t f i r s t I h a d a l s o i n c l u d e d t h e c o n d i t i o n$ $^{'} w i t h o u t {i n d u c t i o n}^{'} b u t i n e d i t i n g t h e q u e s t i o n t h i s$ $h a s b e e n d e l e t e d . N o w I h a v e p o s t e d t h e q u e s t i o n$ $a g a i n w i t h t h e m e n t i o n e d c o n d i t i o n .$
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