Chapter 9.1, Problem 64PPSE

Solution

a.

To write four terms of a sequence that can be described recursively and explicitly.

The first four terms of the sequence are 1, 3, 5 and 7.

Since there is no more instruction given for the question, one can take any sequence.

Consider the sequence $1,3,5,7,9,\mathrm{....}$ .

The first four terms of the sequence are 1, 3, 5 and 7.

b.

To write recursive definition and explicit formula for the sequence.

The recursive definition for the sequence is $a_n=a_{n-1}+2$ , where $a_1=1$ , and the explicit formula for the sequence is $a_n=2n-1$ .

Observe that starting from the second term, each term of the sequence can be obtained by adding 2 to its previous term. Therefore, the recursive definition for the sequence is $a_n=a_{n-1}+2$ , where $a_1=1$ .

Now, observe that the sequence is arithmetic with $a_1=1$ and $d=2$ .

Write the formula for the nth term.

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =1+\left(n-1\right)2\\ =2n-1\end{array}$

Therefore, the explicit formula for the sequence is $a_n=2n-1$ .

b.

To write recursive definition and explicit formula for the sequence.

The recursive definition for the sequence is $a_n=a_{n-1}+2$ , where $a_1=1$ , and the explicit formula for the sequence is $a_n=2n-1$ .

Observe that starting from the second term, each term of the sequence can be obtained by adding 2 to its previous term. Therefore, the recursive definition for the sequence is $a_n=a_{n-1}+2$ , where $a_1=1$ .

Now, observe that the sequence is arithmetic with $a_1=1$ and $d=2$ .

Write the formula for the nth term.

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =1+\left(n-1\right)2\\ =2n-1\end{array}$

Therefore, the explicit formula for the sequence is $a_n=2n-1$ .

c.

To find the 20^th term of the sequence.

Therefore, the 20^th term of the sequence is 39.

The explicit formula for the sequence is $a_n=2n-1$ .

Substitute $n=20$ to find the 20^th term of the sequence.

  $\begin{array}{c}{a}_n=2n-1\\ =2\cdot 20-2\\ =40-1\\ =39\end{array}$

Therefore, the 20^th term of the sequence is 39.