Question

Add the numbers starting from 1 up to 100.then what do u get.
Ex.1+2+3+4+5+6.................100,
what do u get ???????

Answer 1

use the following to find the sum:
\[1+2+3+\cdots +n=\frac{n(n+1)}{2}\]

Answer 2

Let \(S = 1+2+3+4+\cdots+100\)


\[~~1~~+~2~+~3~+~4~+\cdots+100\\~\\100+99+98+97+\cdots+~~1\]

\[2S = \underbrace{101+101+101+\cdots+101}_{\Large100~times} = 100\cdot101\]

So \(S = \dfrac{100\cdot101}{2} = \cdots~?\)

Answer 3

or in general
\[S_n=1+2+3+\cdots+(n-2) +(n-1)+n \\S_n=n+(n-1)+(n-2)+\cdots+3+2+1\\ S_n+S_n \\ =(n+1)+(n-1+2)+(n-2+3)+\cdots +(n-2+3)+(n-1+2)+(n+1) \\ 2 S_n=(n+1)+(n+1)+(n+1) \cdots +(n+1)+(n+1)+(n+1) \\ 2S_n=n(n+1) \\ S_n=\frac{n(n+1)}{2}\]
\[1+2+3+\cdots +n =\frac{n(n+1)}{2} \]