Finding a formula for a difficult sequence

  • Thread starter Thread starter torquerotates
  • Start date Start date
  • Tags Tags
    Formula Sequence
torquerotates
Messages
207
Reaction score
0

Homework Statement

I'm trying to find a formula for 1, 3, 6, 10 ... for n=2,3,4,...







The Attempt at a Solution


Well I know that the difference between each number is increasing by 1. For example, the difference between 1 and 3 is 2, between 3 and 6 is 3, between 6 and 10 is 4 and so on. But, I just don't see a formula.
 
Physics news on Phys.org
Hi! a_n = \sum_{ k = 1 }^n k = \frac{ n ( n + 1 ) }{ 2 }
 
For some reason, I can't read that.
 
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...
Back
Top