How do you prove this by induction?

Question

Prove the following claim holds ∀n ∈ Z≥0 by induction on

Answer 1

Well, we can prove that

∑_0→n i³ = (n⁴ + 2n³ + n²) / 4

(Note that the last term is different from that of the question, which has a typo!)

But first we express the right-hand side of the equation differently by factoring it (because factored forms are likely to be easier to manipulate algebraically). Thus,

∑_0→n i³ = (n⁴ + 2n³ + n²) / 4

= n²(n² + 2n + 1) / 4

= n²(n + 1)² / 4

Thus, we instead prove the following.

∑_0→n i³ = n²(n + 1)² / 4

Proof by Mathematical Induction

Base case. For the case of n = 0, the left-hand side of the equation is ∑_0→n i³ = 0³ = 0, and the right-hand side is n²(n + 1)² / 4 = 0²(0 + 1)² / 4 = 0 / 4 = 0, which makes the equation a true statement.

Inductive step. For the case of n = k + 1, the left-hand side is

∑_0→k+1 i³ = (k+1)³ + ∑_0→k i³

in which we can substitute the inductive hypothesis ∑_0→k i³ = k²(k + 1)² / 4. Thus,

∑_0→k+1 i³ = (k+1)³ + k²(k + 1)² / 4

= (4(k+1)³ + k²(k + 1)²) / 4

= (k+1)²(4(k + 1) + k²) / 4

= (k+1)²(k² + 4k + 4) / 4

= (k+1)²(k + 2)² / 4

= (k+1)²((k+1) + 1)² / 4

which completes the inductive step and the induction proof. 〈