Chapter 12.9, Problem 24E

a.

To determine

To find the next figure in the pattern.

Explanation of Solution

Given:

The given pattern of figures is −

  

Calculation:

The following will be the next figure in the pattern-

  

b.

To determine

To write a sequence to represent the number of dots.

Answer to Problem 24E

  $1,4,9,16,25$

Explanation of Solution

The following is the sequence of dots-

  $\begin{array}{l}{1}^2,2^2,3^2,4^2,5^2,\mathrm{...........}\\ =1,4,9,16,25\end{array}$

c.

To determine

To write the general term for nth term of the sequence in part b.

Answer to Problem 24E

  $a_n=\frac{(n)(n+1)(2n+1)}{6}$

Explanation of Solution

The following is the general term for nth term

  $\begin{array}{l}{1}^2+2^2+3^2+4^2+5^2+\mathrm{...........}+n^2\\ a_n=\frac{(n)(n+1)(2n+1)}{6}\end{array}$

d.

To determine

To write the formula for calculating the total number of dots.

Answer to Problem 24E

  $S_n=\frac{(n)(n+1)(2n+1)}{6}$

Explanation of Solution

The following is formula for calculating the total number of dots-

  $S_n=\frac{(n)(n+1)(2n+1)}{6}$

e.

To determine

To prove the formula obtained in part (d) using mathematical induction.

Explanation of Solution

The following is formula for calculating the total number of dots-

  $S_n=\frac{(n)(n+1)(2n+1)}{6}$

Step 1: Base step

  $S_1=\frac{(1)(1+1)(2(1)+1)}{6}=1$

Step 2:

  $S_k=\frac{(k)(k+1)(2k+1)}{6}$

Step 3: substitute $k=k+1$

  $\begin{array}{l}{S}_{k+1}=\frac{(k+1)(k+2)(2k+2+1)}{6}\\ S_{k+1}=\frac{(k+1)(k+2)(2k+3)}{6}\end{array}$