Question

Use mathematical induction to show that 1^3 + 3^3 + 5^3 + ···+ (2n + 1)^3 = (n + 1)^2(2n^2 + 4n + 1) whenever n is a positive integer.

Answer 1

The question requires us to prove a mathematical statement using induction, showing that the sequence of the cubes of odd numbers equals a given algebraic expression for any positive integer.

Step-by-step explanation:

The student is asking us to show by mathematical induction that the sum of cubes of consecutive odd numbers up to (2n + 1) is equal to (n + 1)2(2n2 + 4n + 1) for any positive integer n.

Base Case (n=1)

For n = 1, the left side of the equation is 13 and the right side is (1 + 1)2(2(1)2 + 4(1) + 1), both simplifying to 4, which confirms our base case.

Inductive Step

Assuming the formula holds for n = k, we test it for n = k + 1. We add (2(k + 1) + 1)3 to both sides of the original equation and simplify to show that it equates to (k + 2)2(2(k + 1)2 + 4(k + 1) + 1), thus completing the induction.