Explain mathematical induction using a simple example not given in the book or lecture notes or the Internet

Explain mathematical induction using a simple example not given in the book or lecture notes or the Internet. [5 points] 2 J for any positive integer n. Let P(n) be the statement 13 + 23 + 33 + + n3 is equal to ("012”) Prove this using mathematical induction (which is done in parts below). [6 points] (a) What is the basis step? Prove this. [1 point] (b) What is the inductive hypothesis? [1 point] (c) What is the inductive step? Prove this, and state your assumptions. [4 points] (d) Explain why these steps show that this formula is true for positive integers. [2 points] . . . 1 1 l 1 1 Consrder the summatlon serles —+—+—+—+...+ 1x2 2X3 3X4 4x5 n(n+1) (a) Find a formula for by examining the values of this expression by summing initial elements of the sequences and ?nding a pattern and form a conjecture. Show your calculation to arrive at the conjecture [4 points] (b) Prove the formula you conjectured in part (a) is correct using mathematical induction. [4 points] Prove that 3" < n! if n is an integer greater than 6. [5 points] Prove using mathematical induction that 3 divides n3 + 2n whenever n is a positive integer. [6 points]