In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n. 1/(1 · 2) + 1/(2 · 3) + 1/(3 · 4) + ... + 1/(n(n+1)) = n/(n + 1)

Verified step by step guidance

1. Step 1: Understand the statement to prove by induction. We want to prove that for every positive integer $n$, the sum $\backslash frac\{1\}\{1\; \backslash cdot\; 2\}\; +\; \backslash frac\{1\}\{2\; \backslash cdot\; 3\}\; +\; \backslash frac\{1\}\{3\; \backslash cdot\; 4\}\; +\; \backslash cdots\; +\; \backslash frac\{1\}\{n(n+1)\}$ equals $\backslash frac\{n\}\{n+1\}$.
2. Step 2: Base Case - Verify the statement for $n=1$. Substitute $n=1$ into the left-hand side (LHS) and right-hand side (RHS) of the equation and check if they are equal.
3. Step 3: Inductive Hypothesis - Assume the statement is true for some positive integer $k$, that is, assume $\backslash frac\{1\}\{1\; \backslash cdot\; 2\}\; +\; \backslash frac\{1\}\{2\; \backslash cdot\; 3\}\; +\; \backslash cdots\; +\; \backslash frac\{1\}\{k(k+1)\}\; =\; \backslash frac\{k\}\{k+1\}$.
4. Step 4: Inductive Step - Using the inductive hypothesis, prove the statement is true for $k+1$. Start with the sum up to $k+1$: $\backslash frac\{1\}\{1\; \backslash cdot\; 2\}\; +\; \backslash frac\{1\}\{2\; \backslash cdot\; 3\}\; +\; \backslash cdots\; +\; \backslash frac\{1\}\{k(k+1)\}\; +\; \backslash frac\{1\}\{(k+1)(k+2)\}$. Replace the sum up to $k$ using the inductive hypothesis.
5. Step 5: Simplify the expression obtained in Step 4 to show it equals $\backslash frac\{k+1\}\{k+2\}$, which completes the inductive step and proves the statement for all positive integers $n$.

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to establish that a statement holds for all positive integers. It involves two steps: proving the base case (usually n=1) is true, and then proving that if the statement holds for an arbitrary integer k, it also holds for k+1. This creates a chain of truth for all n.

Telescoping Series

A telescoping series is a sum where many terms cancel out when expanded, simplifying the expression significantly. In this problem, the sum of fractions can be rewritten to reveal cancellation between consecutive terms, making it easier to find a closed-form expression for the sum.

Algebraic Manipulation of Rational Expressions

Algebraic manipulation involves rewriting fractions and expressions to simplify or transform them. Here, expressing 1/(n(n+1)) as a difference of two fractions (partial fractions) helps in identifying the telescoping pattern and facilitates the induction proof.