Talamancoeb

2022-01-05

I really can't remember (if I have ever known this): which series is this and how to demonstrate its solution?
$\sum _{i=1}^{n}i=\frac{n\left(n+1\right)}{2}$

Anzante2m

Write out this sum twice, once is direct order, and once in reverse:
$1+2+\dots +\left(n-1\right)+n=s$
$n+\left(n-1\right)+\dots +2+1=s$
$\left(n+1\right)+\left(n+1\right)+\dots +\left(n+1\right)+\left(n+1\right)=2s$
There are exactly n terms here (as many as the number of terms in the sum). Hence:
$n\left(n+1\right)=2s$
Now solve for s.

Jonathan Burroughs

This is an Arithmetic Series starting from 1 with difference 1.
$\sum _{i=1}^{n}i=1+2+3+4+\dots +n=\frac{n\left(n+1\right)}{2}$

karton

This is not a series. This sum is named Gauss sum and that formula $\frac{n\left(n+1\right)}{2}$ you can prove it using induction.
The exercise starts from the following sum: 1+2+...+100 and the way you can classify the terms of this sum.
1+2+...+100=(1+100)+(2+99)+...(50+51)

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