What is the formula for calculating the sum of consecutive integers?

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Homework Statement



1+2+3...+n


Homework Equations



count the sum.


The Attempt at a Solution



it's such an easy sum to count but i just want to make sure.

S_{n}=\frac{n}{2}(a_{1}+a_{n})


S_{n}=\frac{n}{2}(1+n)


S_{n}=\frac{n+n^{2}}{2}
 
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Correct. Hope your anxiety is eased
 
Lol its too simple idk why my teacher would give us something so simple I thought there would be a trick or something.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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