Question

Which of the following graphs of exponential functions corresponds to a geometric sequence with a first term of 4 and a ratio of 1/2?

Answer 1

"Graph E" is the one among the following choices given in the question that graphs of exponential functions corresponds to a geometric sequence with a first term of 4 and a ratio of 1/2. The correct option among all the options that are given is the fifth option. I hope the answer has actually come to your help.

Answer 2

Answer: graph E.

A geometric sequence can be written as:

$a_{n} = a_{1} \cdot r^{(n - 1)}$

where:

a₁ = first term = 4

r = ratio = 0.5

Substituting the numbers, we have:

$a_{n} = 4 \cdot (\frac{1}{2})^{n-1}$

or else

$f(x) = 4 \cdot (\frac{1}{2})^{x - 1}$

This is an exponential function with base less than 1. Therefore, we can exclude graph C (which depicts a linear function), and graphs A and D (which depict an exponential function with base greater than 1).

In order to choose between graph B and E, let's evaluate the function in two different points:

$f(1) = 4 \cdot (\frac{1}{2})^{1 - 1} = 4$

$f(2) = 4 \cdot (\frac{1}{2})^{2 - 1} = 4 \cdot \frac{1}{2} = 2$

Therefore, we need to look for the graph passing through the points (1, 4) and (2, 2). That is graph E.