Chapter 9, Problem 9PS

To determine

Prove that the $n^{th}$ pentagonal number $P_n$ .

Answer to Problem 9PS

By mathematical induction the formula is valid for all integers $n\ge 1$ .

Explanation of Solution

Calculation:

Let us try to prove a formula by mathematical induction

Numbers $\text{1},\text{ 5},\text{ 12},\text{ 22},\text{ 35},\text{ 51},\mathrm{.....}$ are pentagonal numbers. They represents the number of dots needed to draw a pentagon as shown below

  

Numbers $\text{1},\text{ 5},\text{ 12},\text{ 22},\text{ 35},\text{ 51},\mathrm{.....}$ can be written recursively as

  $P_n=P_{n-1}+\left(3n-2\right)$

Mathematical induction consists of two distinct parts.

First you must show that formula is true, when $n=1$ , that is,

  $\begin{array}{l}{S}_1=3\left(1\right)-2\\ S_1=1\end{array}$

Assume that

  $P_k=\frac{k\left(3k-1\right)}{2}$ then,

  $\begin{array}{l}{P}_{k+1}=P_k+[3\left(k+1\right)-2]\\ =P_k+(3k+1)\\ =\frac{k\left(3k-1\right)}{2}+\left(3k+1\right)\\ =\frac{3k^2+5k+2}{2}\\ =\frac{\left(k+1\right)\left(3k+2\right)}{2}\\ =\frac{\left(k+1\right)[3\left(k+1\right)-1]}{2}\end{array}$

Hence, by mathematical induction the formula is valid for all integers $n\ge 1$ .