Problems

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 $n$.

Illustrate with a picture

(a) $(a-b{)}^2=a^2-2ab+b^2$,

(b) $a^2-b^2=(a-b)(a+b)$,

(c) $(a+b+c{)}^2=a^2+b^2+c^2+2ab+2ac+2bc$.

Suppose $a>b$. Explain using the number line why

(a) $a-c>b-c$, (b) $2a>2b$.

Suppose $a>b$ and $c>d$. Prove that $a+c>b+d$.

Using mathematical induction prove that $2^n\ge n+1$ for all natural numbers.