To prove the given statement by induction:

Statement:

$\sum_{k=1}^n \frac{1}{(2k-1)(2k+1)} = \frac{n}{2n+1}.$

Step 1: Base Case

For $n = 1$, the left-hand side is: $\frac{1}{(2 \cdot 1 - 1)(2 \cdot 1 + 1)} = \frac{1}{1 \cdot 3} = \frac{1}{3}.$ The right-hand side is: $\frac{1}{2 \cdot 1 + 1} = \frac{1}{3}.$ Since both sides are equal, the base case holds true.

Step 2: Inductive Hypothesis

Assume the statement holds for some $n = k$: $\sum_{i=1}^k \frac{1}{(2i-1)(2i+1)} = \frac{k}{2k+1}.$

Step 3: Inductive Step

Prove the statement holds for $n = k+1$. The left-hand side for $n = k+1$ becomes: $\sum_{i=1}^{k+1} \frac{1}{(2i-1)(2i+1)} = \sum_{i=1}^k \frac{1}{(2i-1)(2i+1)} + \frac{1}{(2(k+1)-1)(2(k+1)+1)}.$ Using the inductive hypothesis: $\sum_{i=1}^k \frac{1}{(2i-1)(2i+1)} = \frac{k}{2k+1}.$ Thus: $\sum_{i=1}^{k+1} \frac{1}{(2i-1)(2i+1)} = \frac{k}{2k+1} + \frac{1}{(2(k+1)-1)(2(k+1)+1)}.$

Simplify the second term: $\frac{1}{(2(k+1)-1)(2(k+1)+1)} = \frac{1}{(2k+1)(2k+3)}.$

Now combine the terms: $\frac{k}{2k+1} + \frac{1}{(2k+1)(2k+3)} = \frac{k(2k+3) + 1}{(2k+1)(2k+3)}.$

Simplify the numerator: $k(2k+3) + 1 = 2k^2 + 3k + 1.$

So: $\frac{k}{2k+1} + \frac{1}{(2k+1)(2k+3)} = \frac{2k^2 + 3k + 1}{(2k+1)(2k+3)}.$

Factor the numerator: $2k^2 + 3k + 1 = (k+1)(2k+1).$

Thus: $\frac{2k^2 + 3k + 1}{(2k+1)(2k+3)} = \frac{(k+1)}{2k+3}.$

This is the right-hand side for $n = k+1$: $\frac{k+1}{2(k+1)+1}.$

Conclusion

By induction, the statement is true for all natural numbers $n$.

Let me know if you'd like detailed explanations on any step or related concepts! Here are 5 related questions to explore further:

1. Can you generalize the proof for a broader class of telescoping series?
2. How do telescoping sums simplify proofs involving series?
3. What happens if the summand has a higher-order denominator?
4. How does this proof change for non-natural numbers (e.g., fractions)?
5. Can this series be solved using alternative methods, such as partial fractions?

Tip: Always verify the telescoping nature of a series by analyzing its