Chapter 9.4, Problem 31E

To determine

To proof: The expression $n^3+3n^2+2n$ is a factor of 3 with help of mathematical induction for integers $n\ge 1$ .

Explanation of Solution

Given information:

The expression $n^3+3n^2+2n$ .

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 $n^3+3n^2+2n$ .

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,

  $\begin{array}{c}{1}^3+3{\left(1\right)}^2+2\left(1\right)=1+3+2\\ =6\end{array}$

The above equation holds true as 3 is a factor of 6.

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

Now, assume that the statement if true for $n=k$ so, $k^3+3k^2+2k$ is a factor of 3.

Let $k^3+3k^2+2k=3a$ where a is any number.

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

Now, prove it for $n=k+1$ . Write the expression for $n=k+1$ .

  $\begin{array}{c}{\left(k+1\right)}^3+3{\left(k+1\right)}^2+2\left(k+1\right)=k^3+1+3k^2+3k+3k^2+3+6k+2k+2\\ =k^3+3k^2+2k+\left(1+3k^2+9k+2\right)\\ =k^3+3k^2+2k+\left(3+3k^2+9k\right)\end{array}$

Substitute the value of $S_k$ ,

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

Where $\lambda =\left(a+1+k^2+3k\right)$

Therefore, it is concluded that 3 is a factor of ${\left(k+1\right)}^3+3{\left(k+1\right)}^2+2\left(k+1\right)$ where $\left(a+1+k^2+3k\right)$ is any polynomial.

Hence, by the principle of mathematical induction the statement $n^3+3n^2+2n$ is a factor of 3 is true for all integers $n\ge 1$ .