Chapter 7.8, Problem 2P

a.

To determine

To write: a recursive formula and an explicit formula for the given sequence.

Answer to Problem 2P

  $\begin{array}{l}g(x)=g(x-1)\times 2\\ $ g(x)=2{(2)}^{x-1}$\end{array}$

Explanation of Solution

Given information: a sequence is given as follows-

2, 4, 8, 16 . . .

Here the ratio $\frac{4}{2}=\frac{8}{4}=\frac{16}{8}=2$ is same

So, the given sequence is in geometric progression whose common ratio is 2 and first term is 2.

Then $x^{th}$ term is given by $g(x)=2{(2)}^{x-1}$ , this is the explicit formula.

Now every term is obtained by multiplying by 2 in the preceding term, so the recursive formula of given geometric sequence is given by $g(x)=g(x-1)\times 2$ .

b.

To determine

To write: a recursive formula and an explicit formula for the given sequence.

Answer to Problem 2P

  $\begin{array}{l}g(x)=g(x-1)\times \frac{1}{2}\\ g(x)=40{\left(\frac{1}{2}\right)}^{x-1}\end{array}$

Explanation of Solution

Given information: a sequence is given as follows-

40, 20, 10, 5 . . .

Here the ratio $\frac{40}{20}=\frac{10}{20}=\frac{5}{10}=\frac{1}{2}$ is same

So, the given sequence is in geometric progression whose common ratio is $\frac{1}{2}$ and first term is 40.

Then $x^{th}$ term is given by $g(x)=40{\left(\frac{1}{2}\right)}^{x-1}$

Now every term is obtained by multiplying by $\frac{1}{2}$ in the preceding term, so the recursive formula of given geometric sequence is given by $g(x)=g(x-1)\times \frac{1}{2}$ .