Induction Proof Help: Showing (n^5)/5 + (n^3)/3 + 7n/15 is Integer

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I need to show that (n^5)/5 + (n^3)/3 + 7n/15 is an integer for all n.


I tried induction that obviously work for 1 but i could not manage to show this for k+1. Any tips please?
 
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try expanding the k+1 case. Pascals triangle is a huge help for figuring out how to expand it. After expanding and factoring, you should see the base case and your hypothesis.
 

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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