Without-using-arithmatic-Series-concept-or-formula-prove-the-following-1-2-3-n-n-n-1-2-

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 + \dots + 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, \dots .

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 .