Chapter 11.4, Problem 2E

To determine

To prove:

Given statement is true using principle of mathematical induction.

Explanation of Solution

Give information:

  $\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+\mathrm{...}+\frac{1}{\left(3n-1\right)\left(3n+1\right)}=\frac{n}{3n+1}$

Formula Used:

Principle of mathematical induction.

Proof:

It is given that,

  $\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+\mathrm{...}+\frac{1}{\left(3n-1\right)\left(3n+1\right)}=\frac{n}{3n+1}$

By using principle of mathematical induction,

  $LHS=RHS$

Let’s take, left hand side,

  $\begin{array}{l}LHS=\frac{1}{1.4}\\ LHS=\frac{1}{4}\end{array}$

Now take $RHS$ (right hand side),

  $\begin{array}{l}RHS=\frac{n}{\left(3n+1\right)}\\ RHS=\frac{1}{\left(3.1+1\right)}\\ RHS=\frac{1}{\left(3+1\right)}\\ RHS=\frac{1}{4}\end{array}$

Therefore, $LHS=RHS$

Therefore, given statement is true.