Chapter 9.4, Problem 11E

To determine

To proof: The expression $2+4+6+8+\cdots +2n=n\left(n+1\right)$ with help of mathematical induction for integers $n\ge 1$ .

Explanation of Solution

Given information:

The expression $2+4+6+8+\cdots +2n=n\left(n+1\right)$ .

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 $2+4+6+8+\cdots +2n=n\left(n+1\right)$ .

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,

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

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 $n=k$ .

  $2+4+6+8+\cdots +2k=k\left(k+1\right)$  ...(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 $S_k$ .

  $2+4+6+8+\cdots +2k+2\left(k+1\right)=S_k+2\left(k+1\right)$

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

  $\begin{array}{c}2+4+6+8+\cdots +2k+2\left(k+1\right)=S_k+2\left(k+1\right)\\ =k\left(k+1\right)+2\left(k+1\right)\end{array}$

Factor out the terms and simplify,

  $\begin{array}{c}2+4+6+8+\cdots +2k+2\left(k+1\right)=S_k+2\left(k+1\right)\\ =k\left(k+1\right)+2\left(k+1\right)\\ =\left(k+1\right)\left(k+2\right)\\ =\left(k+1\right)\left(k+1+1\right)\end{array}$

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

  $2+4+6+8+\cdots +2k+2\left(k+1\right)=\left(k+1\right)\left(k+1+1\right)$

Hence, by the principle of mathematical induction the statement $2+4+6+8+\cdots +2n=n\left(n+1\right)$ is true for all integers $n\ge 1$ .