Chapter 9, Problem 65RE

To determine

To proof: The expression $3+5+7+\cdots +\left(2n+1\right)=n\left(n+2\right)$ with help of mathematical induction for integers $n\ge 1$ .

Explanation of Solution

Given information:

The expression $3+5+7+\cdots +\left(2n+1\right)=n\left(n+2\right)$ .

Formula used:

The steps to prove a statement by mathematical induction are as follows:

Step 1. Show that the statement is true for $n=1$ .

Step 2. Assume that the statement is true for $n=k$ .

Step 3. Prove that the statement is true for $n=k+1$ with help of step 2.

Proof:

Consider expression $3+5+7+\cdots +\left(2n+1\right)=n\left(n+2\right)$ .

Denote the expression by $S_n$ .

Recall that the steps to prove a statement by mathematical induction are as follows:

Step 1. Show that the statement is true for $n=1$ .

Step 2. Assume that the statement is true for $n=k$ .

Step 3. Prove that the statement is true for $n=k+1$ with help of step 2.

Substitute $n=1$ in the expression,

  $\begin{array}{c}2\left(1\right)+1=1\left(1+2\right)\\ 3=3\end{array}$

The above equation holds true.

Therefore, the statement is true for $n=1$ so $S_1$ is true.

Now, assume that the statement if true for $n=k$ .

  $3+5+7+\cdots +\left(2k+1\right)=k\left(k+2\right)$

  ...(1)

Therefore, the statement holds true for $n=k$ so $S_k$ is true.

Now, prove it for $n=k+1$ . Write the expression for $n=k+1$ . Group the terms that involve $S_k$ .

  $3+5+7+\cdots +\left(2k+1\right)+\left[2\left(k+1\right)+1\right]=S_k+\left[2\left(k+1\right)+1\right]$

Substitute the value of $S_k$ from equation (1),

  $\begin{array}{c}3+5+7+\cdots +\left(2k+1\right)+\left[2\left(k+1\right)+1\right]=S_k+\left[2\left(k+1\right)+1\right]\\ =k\left(k+2\right)+\left[2\left(k+1\right)+1\right]\\ =k^2+2k+2k+3\\ =k^2+4k+3\end{array}$

Factor out the terms and simplify,

  $\begin{array}{c}3+5+7+\cdots +\left(2k+1\right)+\left[2\left(k+1\right)+1\right]=k^2+3k+k+3\\ =k\left(k+3\right)+1\left(k+3\right)\\ =\left(k+1\right)\left(k+3\right)\\ =\left(k+1\right)\left[\left(k+1\right)+2\right]\end{array}$

Result obtained is that $S_k$ implies $S_{k+1}$ that is,

  $3+5+7+\cdots +\left(2k+1\right)+\left[2\left(k+1\right)+1\right]=\left(k+1\right)\left[\left(k+1\right)+2\right]$

Hence, by the principle of mathematical induction the statement $3+5+7+\cdots +\left(2n+1\right)=n\left(n+2\right)$ is true for all integers $n\ge 1$ .