Chapter 9.5, Problem 13E

Solution

To prove : using mathematical induction to prove that statement $1+5+9+\cdots +\left(4n-3\right)=n\left(2n-1\right)$ holds true for all positive integer.

Given information :

The statement $1+5+9+\cdots +\left(4n-3\right)=n\left(2n-1\right)$ .

Formula used :

Follow the steps to prove that the statement is true for any positive integer.

Anchor step: Prove that any statement $P_n$ is true for $n=1$ .

Inductive hypothesis: Assume $P_n$ is true for $n=k$ .

Inductive Step: Prove that $P_n$ is true for $n=k+1$ .

Proof :

Conjecture:

Take $P_n$ to be statement $1+5+9+\cdots +\left(4n-3\right)=n\left(2n-1\right)$ .

Anchor step:

Prove that $P_n$ is true for $n=1$ .

For $n=1$ ,

  $\begin{array}{c}4\left(1\right)-3=1\left(2\left(1\right)-1\right)\\ 4-3=1\left(2-1\right)\\ 1=1\end{array}$

Inductive hypothesis:

Assume $P_n$ is true for $n=k$ . Replace $n$ with $k$ in $P_n$ .

  $1+5+9+\cdots +\left(4k-3\right)=k\left(2k-1\right)$

Inductive Step:

Prove that $P_n$ is true for $n=k+1$ . This means it is needed to be proved that $P_{k+1}$ must be true.

Proceed with the inductive hypothesis.

  $1+5+9+\cdots +\left(4k-3\right)=k\left(2k-1\right)$

Add ${\left(k+1\right)}^{th}$ term on both sides and simplify the right-hand side.

  $\begin{array}{c}1+5+9+\cdots +\left(4k-3\right)+\left[4\left(k+1\right)-3\right]=k\left(2k-1\right)+\left[4\left(k+1\right)-3\right]\\ =2k^2-k+4k+4-3\\ =2k^2+3k+1\end{array}$

Factor the right-hand side.

  $\begin{array}{c}1+5+9+\cdots +\left(4k-3\right)+\left[4\left(k+1\right)-3\right]=\left(k+1\right)\left(2k+1\right)\\ =\left(k+1\right)\left[2\left(k+1\right)-1\right]\end{array}$

The statement is same as the statement $P_{k+1}$ , so $P_{k+1}$ is true.

So, if $P_n$ is true for $n=k$ , then it is also true for $n=k+1$ .

Conclusion:

As $P_n$ is true for $n=1$ and $P_k$ further implies $P_{k+1}$ , this means $P_n$ is true for $n=2$ , $n=3$ and so on.

By mathematical induction, $P_n:1+5+9+\cdots +\left(4n-3\right)=n\left(2n-1\right)$ is true for all positive integers $n$ .