Mathematical induction - exercises

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

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.

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

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 .)