Use mathematical induction in Exercises 3-17 to prove...

Use mathematical induction in Exercises $3-17$ to prove summation formulae. Be sure to identify where you use the inductive hypothesis. a) Find a formula for $$ \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\cdots+\frac{1}{n(n+1)} $$ by examining the values of this expression for small values of $n .$ b) Prove the formula you conjectured in part (a).

Use mathematical induction in Exercises $3-17$ to prove summation formulae. Be sure to identify where you use the inductive hypothesis.
a) Find a formula for
$$
\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\cdots+\frac{1}{n(n+1)}
$$
by examining the values of this expression for small values of $n .$
b) Prove the formula you conjectured in part (a).

Key Concepts

Mathematical Induction

Mathematical induction is a method used to prove statements about natural numbers. It involves two main steps: verifying the base case (usually the smallest natural number) and then proving that if the statement holds for an arbitrary natural number k, it must also hold for k + 1. This systematic approach assures that the statement is true for all natural numbers.

Base Case

The base case is the starting point of an induction proof. It involves demonstrating that the statement is true for the initial value, often n = 1. This step provides the foundation upon which the inductive argument is built.

Inductive Hypothesis

The inductive hypothesis is the assumption that the statement is true for a particular natural number, say n = k. This assumption is critical as it serves as the basis for proving the statement for the next number (n = k + 1) in the inductive step.

Inductive Step

The inductive step involves proving that if the statement holds for an arbitrary natural number k (as assumed by the inductive hypothesis), then it must also hold for the next number, k + 1. This step is essential to extend the truth of the statement to all natural numbers.

Telescoping Series

A telescoping series is one in which most terms cancel out when the series is expanded, leaving only a few terms that determine the sum. Recognizing a telescoping pattern is particularly useful in summation problems because it allows for a significant simplification of the expression, often leading directly to a closed-form formula.

Transcript

00:01 Hi, in this question we are asked to figure out the formula for this sum over here and then prove that the formula we found is true for any positive integer n.

00:16 So we are using induction again.

00:19 First, what should the formula for this sum be? well, you can look at each term here and see that it can can be broken into two terms always.

00:39 So it will be one over the first number minus one over the second number, always.

00:50 So until the end here, it will be one over n minus one over n plus one, right? and everything in between gonna queue each other out left with just the first and the last term.

01:09 So this should be our formula that we found.

01:13 So to answer the first question, this is the formula for this sum.

01:21 And we want to prove that it is true for any integer n, positive integer n.

01:31 So we are using mat induction.

01:34 First, basic step.

01:37 Oh, sorry.

01:38 Let's call this formula p of n.

01:43 Okay, and basic step p of one.

01:46 So the left hand side will have one terms in the sum.

01:51 The right hand side is this one, and it is one over two.

01:56 So p1 saying that 1 over 2 equal itself, which is obviously true.

02:05 So we have clear the basic step.