Chapter 9, Problem 10PS

(a)

To determine

To find:The conclusion from the statement that $P_3$ is true and $P_k$ implies $P_{k+1}$ .

Answer to Problem 10PS

The conclusion is given below.

Explanation of Solution

Calculation:

The formula $P_n=\frac{n\left(3n-1\right)}{2}$ represents the nth pentagonal number for all values of $n\ge 1$ .

Given that $P_3$ is true, this implies that here the value of n starts from 3 and $P_k$ implies $P_{k+1}$ means that if the formula holds for k , then it holds for k +1 as well.

Therefore, the statement implies that $P_n$ holds true for all values of n , $n\ge 3$ .

(b)

To determine

To find:The conclusion from the statement that $P_1,P_2,P_3,\dots ,P_{50}$ are all true.

Answer to Problem 10PS

The conclusion is given below.

Explanation of Solution

Calculation:

The formula $P_n=\frac{n\left(3n-1\right)}{2}$ represents the nth pentagonal number for all values of $n\ge 1$ .

Given that $P_1,P_2,P_3,\dots ,P_{50}$ is true, this implies that $P_n$ holds true for all values of n , $1\le n\le 50$ .

(c)

To determine

To find:The conclusion from the statement that $P_1$ , $P_2$ and $P_3$ are all true, but the truth of $P_k$ does not imply that $P_{k+1}$ is true.

Answer to Problem 10PS

The conclusion is given below.

Explanation of Solution

Calculation:

The formula $P_n=\frac{n\left(3n-1\right)}{2}$ represents the nth pentagonal number for all values of $n\ge 1$ .

Given that $P_1$ , $P_2$ and $P_3$ are all true, this implies that $P_n$ holds true for $n=1,2,3$ and truth of $P_k$ does not imply $P_{k+1}$ is true means that the truth of $P_1$ , $P_2$ and $P_3$ does not imply the truth of $P_4$ , $P_5$ and so on.

Therefore, the statement implies that $P_n$ holds true for only $n=1,2,3$ .

(d)

To determine

To find:The conclusion from the statement that $P_2$ is true and $P_{2k}$ implies $P_{2k+2}$ .

Answer to Problem 10PS

The conclusion is given below.

Explanation of Solution

Calculation:

The formula $P_n=\frac{n\left(3n-1\right)}{2}$ represents the nth pentagonal number for all values of $n\ge 1$ .

Given that $P_3$ is true, this implies that here the value of n starts from 2 and $P_{2k}$ implies $P_{2k+2}$ means that the formula holds for all even integers.

Therefore, the statement implies that $P_n$ holds true for all even integers greater than or equal to 2.