Induction Problems: Solving (n^2-n)/2 & sqrt(n) < 1/sqrt(1)+...+1/sqrt(n)

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Homework Statement


show that for n points in a plane, with no 3 points colinear, the number of line segments joining all pairs of points is (n^2-n)/2


Problem 2
Show that sqrt(n) is strictly less than 1/sqrt(1) +1/sqrt(2)+...+1/sqrt(n) for n\geq 2


The Attempt at a Solution


For problem 1, I have no idea how to start. For problem 2, I tried manipulating the equation and substituting in the induction assumption, but I couldn't get anywhere with it.
 
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Suppose you already have n points all joined up and you add a point somewhere else not joined to any other point. How many lines do you have to draw to connect this one point to every point in the existing diagram?
 
I figured out the first problem, but I'm still having trouble with the second. Any suggestions?
 
You haven't shown us what you did, so we don't know what your problem is. What do you need to prove to make the induction work?
 

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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