Question

Use mathematical induction to prove the statement is true for every positive integer n. 1² 3² 5*2 ... (2n - 1)² = n(2n - 1)(2n + 1) /3?

Answer 1

Using mathematical induction, we start with the base case for n=1 and then assume the formula holds true for n=k. In the inductive step, we show the formula is valid for n=k+1 by adding the next squared term and simplifying, thereby proving that the original statement holds for all positive integers n.

Step-by-step explanation:

We are going to prove by mathematical induction that the sum of the squares of the first n odd numbers equals to n(2n - 1)(2n + 1) / 3. The formula is: 1² + 3² + 5² + ... + (2n - 1)² = n(2n - 1)(2n + 1) / 3.

Base Case: Let's start with n = 1. We have 1² = 1(2*1 - 1)(2*1 + 1) / 3 which simplifies to 1 = 1. Therefore, the base case holds true.

Induction Hypothesis: Assume that the statement is true for n = k, which means 1² + 3² + ... + (2k - 1)² = k(2k - 1)(2k + 1) / 3.

Inductive Step: We need to show that this is true for n = k + 1. So, we add the next term (2(k + 1) - 1)² to both sides of our induction hypothesis and simplify. After simplification, we should arrive at (k + 1)(2(k + 1) - 1)(2(k + 1) + 1) / 3, thereby verifying the induction step is valid and the formula holds true for every positive integer n.