Question

Prove the statement using mathematical induction. 1 ³+2³ +⋯+n³ =[ n(n+1)/2 ]² , for every integer n≥1

Answer 1

Using mathematical induction, we verified the base case for n=1 and then assumed the statement to be true for an integer k. After algebraic manipulation, we showed that the formula holds for k+1, completing the inductive step and proving the statement for all n ≥ 1.

Step-by-step explanation:

We are tasked with proving the statement using mathematical induction: 1³ + 2³ + … + n³ = [n(n+1)/2]² for every integer n ≥ 1. Let's tackle this problem step by step.

1. Base case: For n=1, the left-hand side (LHS) is 1³ = 1, and the right-hand side (RHS) is [(1)(1+1)/2]² = 1. Since LHS = RHS, the base case is true.
2. Inductive step: Assume the statement is true for some integer k, so 1³ + 2³ + … + k³ = [k(k+1)/2]². We need to prove it for k+1.
3. Consider the sum 1³ + 2³ + … + k³ + (k+1)³.
   By the inductive hypothesis, this is [k(k+1)/2]² + (k+1)³.
4. After some algebraic manipulation, this simplifies to [(k+1)(k+2)/2]², which is precisely the formula with n replaced by k+1.

Therefore, by mathematical induction, the statement is proven to be true for every integer n ≥ 1.