Letp(n)bethestatementthat13 +23 +⋯+n3 = (n(n+ 1)∕2)2 for the positive integer n.
a) what is the statement p(1)?
b) show that p(1) is true, completing the basis step of the proof of p(n) for all positive integers n.
c) what is the inductive hypothesis of a proof that p(n) is true for all positive integers n?
d) what do you need to prove in the inductive step of a proof that p(n) is true for all positive integers n?
e) complete the inductive step of a proof that p(n) is true for all positive integers n, identifying where you use the inductive hypothesis.
f) explain why these steps show that this formula is true whenever n is a positive integer.