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?

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

Anzante2m

Beginner2022-01-06Added 34 answers

Write out this sum twice, once is direct order, and once in reverse:

$1+2+\dots +(n-1)+n=s$

$n+(n-1)+\dots +2+1=s$

Now add up column-wise:

$(n+1)+(n+1)+\dots +(n+1)+(n+1)=2s$

There are exactly n terms here (as many as the number of terms in the sum). Hence:

$n(n+1)=2s$

Now solve for s.

Now add up column-wise:

There are exactly n terms here (as many as the number of terms in the sum). Hence:

Now solve for s.

Jonathan Burroughs

Beginner2022-01-07Added 37 answers

This is an Arithmetic Series starting from 1 with difference 1.

$\sum _{i=1}^{n}i=1+2+3+4+\dots +n=\frac{n(n+1)}{2}$

karton

Expert2022-01-11Added 613 answers

This is not a series. This sum is named Gauss sum and that formula

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)

What is the derivative of the work function?

How to use implicit differentiation to find $\frac{dy}{dx}$ given $3{x}^{2}+3{y}^{2}=2$?

How to differentiate $y=\mathrm{log}{x}^{2}$?

The solution of a differential equation y′′+3y′+2y=0 is of the form

A) ${c}_{1}{e}^{x}+{c}_{2}{e}^{2x}$

B) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{3x}$

C) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{-2x}$

D) ${c}_{1}{e}^{-2x}+{c}_{2}{2}^{-x}$How to find instantaneous velocity from a position vs. time graph?

How to implicitly differentiate $\sqrt{xy}=x-2y$?

What is 2xy differentiated implicitly?

How to find the sum of the infinite geometric series given $1+\frac{2}{3}+\frac{4}{9}+...$?

Look at this series: 1.5, 2.3, 3.1, 3.9, ... What number should come next?

A. 4.2

B. 4.4

C. 4.7

D. 5.1What is the derivative of $\frac{x+1}{y}$?

How to find the sum of the infinite geometric series 0.9 + 0.09 + 0.009 +…?

How to find the volume of a cone using an integral?

What is the surface area of the solid created by revolving $f\left(x\right)={e}^{2-x},x\in [1,2]$ around the x axis?

How to differentiate ${x}^{\frac{2}{3}}+{y}^{\frac{2}{3}}=4$?

The differential coefficient of $\mathrm{sec}\left({\mathrm{tan}}^{-1}\left(x\right)\right)$.