Without-using-induction-or-arithmatic-series-concept-prove-the-following-1-2-3-n-n-n-1-2-

Question Number 2762 by Rasheed Soomro last updated on 26/Nov/15

W i t h o u t u s i n g \underline{i n d u c t i o n} o \underline{r a r i t h m a t i c s e r i e s - c o n c e p t}

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 prakash jain last updated on 26/Nov/15

n^{2} - {(n - 1)}^{2} = 2 n - 1

1^{2} - 0^{2} = 2 \cdot 1 - 1

2^{2} - 1^{2} = 2 \cdot 2 - 1

3^{2} - 2^{2} = 3 \cdot 2 - 1

\dots

\dots

n^{2} - {(n - 1)}^{2} = 2 \cdot n - 1

Sum all of the above .

n^{2} = 2 (1 + 2 + . . + n) - n

2 (1 + 2 + 3 + . . + n) = n^{2} + n

\underset{i = 1}{\sum^{n}} i = \frac{n (n + 1)}{2}

Commented by Rasheed Soomro last updated on 27/Nov/15

T h a n k S!