Question

Problem 1. Fill in the blank with a closed-form expression, and then give careful proof by mathematical induction that the resulting equation is true for all integers n≥1.

1/1⋅2+ 1/2⋅3 +⋯+ 1/n(n+1) =

Answer 1

The closed-form expression for the given series 1/1⋅2 + 1/2⋅3 + ⋅ + 1/n(n+1) is 1 - 1/(n+1), and it's proven to be true for all integers n ≥1 using mathematical induction, involving a base case verification and an inductive step.

Step-by-step explanation:

The closed-form expression for the series 1/1⋅2 + 1/2⋅3 + ⋅ + 1/n(n+1) is 1 - 1/(n+1). This can be proven using mathematical induction.

Base Case (n=1)

Consider n=1. The left side is 1/1⋅2, which simplifies to 1/2. The right side using our closed-form expression is 1 - 1/2, which also simplifies to 1/2. Thus, the base case holds.

Inductive Step

Assume the statement is true for some integer k, so 1/1⋅2 + 1/2⋅3 + ⋅ + 1/k(k+1) = 1 - 1/(k+1). Now consider the case when n=k+1. According to the inductive hypothesis, add 1/[(k+1)(k+2)] to both sides, which gives us the new sum 1 - 1/[(k+1)+1], simplifying to 1 - 1/(k+2), confirming the pattern holds for k+1.

Thus, by mathematical induction, the closed-form expression holds for all integers n ≥1.