Give a proof by induction of the following statement. If n is a positive integer, then n Σ(7- k) = = k=1 n(13 − n) 2

| 3. Use induction to prove that n2 – n is always even. -

Use the method of mathematical induction to prove the following statement where n is a positive integer. n(3n+5) 4+7+10+ (3n + 1) -

12n – 5n is divisible by 7, for all n  0. Give the formal proof of one of the following (your choice), using Mathematical Induction.

Use the principle of mathematical induction to show that the statement is true for all natural numbers. 2² +4² +6² + ... + (2n)² = 2n(n + 1)(2n + 1) 3 Let Pn denote the statement: 22 +4² +6² + ... + (2n)² + Check that P₁ is true: 2² = 4 and 2(C Assume Pk is true: 22 +42 +62 + + (² 2 (2n)² = 2n(n + 1)(2n + 1) 3 2² +4² +6² + ... + (2(k+1))² ))( + ¹)(² 1) (2 1 3 To show that Pk+1 is true, add (2(k + 1))² to both sides of Pk. 2² + 4² + 6² + ... + (2k)² + (2( ))²³ = 2 2 |)²-² = +1 1) 2k(k + 1)(2k + 1) 12( + 3 = Rewrite the right-hand side as a single fraction, and then factor the numerator completely. 3 3 2k(k + 1)(2k + 1) +(2( 3 (k + 1)(2k + 1) Thus P₁ is true. 2 2 :))²

Prove the statement is true by using Mathematical Induction.

4. Use mathematical induction to show that: 12 + 22 + 32 + … + n² = (n(n+1)(2n+1)) 6