Chapter 8.5, Problem 69E

a.

To determine

To write an explicit rule for each sequence.

Answer to Problem 69E

  $T_n=T_{n-1}+n$ . $S_n=S_{n-1}+2n-1$

Explanation of Solution

Given:

The first four triangular numbers are $T_n$ and the first four square numbers $S_n$ , are represented by the points in each diagram.

  

Calculation:

The explicit rule for sequence $T_n$ .

  $\begin{array}{l}{T}_1=1\\ T_n=T_{n-1}+n\end{array}$

The explicit rule for sequence $S_n$ .

  $\begin{array}{l}{S}_1=1\\ S_n=S_{n-1}+2n-1\end{array}$

b.

To determine

To write a recursive rule for each sequence.

Answer to Problem 69E

  $a_n=a_{n-1}+2$ , $a_n=2n-1$

Explanation of Solution

Given:

The first four triangular numbers are $T_n$ and the first four square numbers $S_n$ , are represented by the points in each diagram.

  

Calculation:

The following is the recursive rule-

  $a_n=a_{n-1}+2$

And the following is the explicit rule-

  $\begin{array}{l}{a}_n=1+(n-1)2\\ a_n=1+2n-2\\ a_n=2n-1\end{array}$

c.

To determine

To write a rule for square numbers in terms of square numbers.

Answer to Problem 69E

  $a_n=a_{n-1}+2$ , $a_n=2n-1$

Explanation of Solution

Given:

The first four triangular numbers are $T_n$ and the first four square numbers $S_n$ , are represented by the points in each diagram.

  

Calculation:

The following is the recursive rule-

  $a_n=a_{n-1}+2$

And the following is the explicit rule-

  $\begin{array}{l}{a}_n=1+(n-1)2\\ a_n=1+2n-2\\ a_n=2n-1\end{array}$