Problems

Using mathematical induction prove that $$1+2+3+\cdots +n=\frac{n(n+1)}{2}.$$

There are $n$ lines on a plane, and all the lines intersect at exactly one point. Prove that the lines divide the plane into $2n$ parts.

There are $n$ lines on a plane, no two lines are parallel, and no three lines cross at one point. Show that those lines dived the plane into $\frac{n(n+1)}{2}+1$ regions.

In a sequence 2, 6, 12, 20, 30, ... find the number

(a) in the 6th place

(b) in the 2016th place.

Using mathematical induction prove that $$1+3+5+\cdots +(2n-1)=n^2.$$

Circles and lines are drawn on the plane. They divide the plane into non-intersecting regions, see the picture below.

Show that it is possible to colour the regions with two colours in such a way that no two regions sharing some length of border are the same colour.

Numbers $1,2,\dots ,n$ are written on a whiteboard. In one go Louise is allowed to wipe out any two numbers $a$ and $b$, and write their sum $a+b$ instead. Louise enjoys erasing the numbers, and continues the procedure until only one number is left on the whiteboard.

What number is it? What if instead of $a+b$ she writes $a+b-1$?

Prove that

(a) $$1^2+2^2+3^2+\cdots +n^2=\frac{1}{6}n(n+1)(2n+1)$$

(b) $$1^2+3^2+5^2+\cdots +(2n-1{)}^2=\frac{1}{3}n(2n-1)(2n+1).$$

Using mathematical induction prove that $2^n>n$ for all natural numbers.

Using mathematical induction show that $2^n>n$ for all natural numbers $n$.