Mathematical induction - exercises

Exercise.

Prove using induction that =1 2= (+1)(2+1) 6 .

Exercise.

Given a chocolate bar consisting of a number of squares arranged in a rectangular pattern, split the bar into small squares (always breaking along the lines between the squares) with a minimum number of breaks. How many will it take? Prove your answer using induction.

Exercise.

Use the minimal counter-example principle to prove that there are no positive integers , such that 2=2 2 .

Exercise.

Prove that for every positive integer there exists an -digit number divisible by 5 all of whose digits are odd. (Hint: using induction on . If 12 is divisible by 5 show that one of 112 , 312 , 512 , 712 , 912 is divisible by 5 +1 .)