Prove by mathematical induction that11×2×3+12×3×4+ ⋯ +1n(n+1)(n+2)=14−12(n+1)(n+2).

Sure, let's prove the given statement using the principle of mathematical induction.

Mathematical induction has two steps: the base case and the inductive step.

Step 1: Base Case Let's start with n=1. The left side of the equation becomes 11×2×3 = 6, and the right side becomes 1/4 - 1/2(1+1)(1+2) = 1/4 - 1/2(2)(3) = 1/4 - 1/3 = 1/12. So, the base case does not hold for n=1.

Step 2: Inductive Step Assume the statement is true for n=k, i.e., 11×2×3 + 12×3×4 + ... + 1k(k+1)(k+2) = 1/4 - 1/2(k+1)(k+2).

We need to prove the statement is true for n=k+1, i.e., 11×2×3 + 12×3×4 + ... + 1k(k+1)(k+2) + 1(k+1)(k+2)(k+3) = 1/4 - 1/2(k+2)(k+3).

Starting with the left side of the equation for n=k+1, we can substitute the inductive hypothesis:

11×2×3 + 12×3×4 + ... + 1k(k+1)(k+2) + 1(k+1