To find the formula and graph of the sequence defined by the function \( f(x+1) = \frac{2}{3} f(x) \) with an initial value of 108, let's break down the steps and derive the terms of the sequence.
1. Initial Value:
The sequence starts with an initial value of \( f(0) = 108 \).
2. Recursive Formula:
The given recursive formula is \( f(x+1) = \frac{2}{3} f(x) \).
This means that each term in the sequence is obtained by multiplying the previous term by \( \frac{2}{3} \).
3. Finding the First Few Terms:
Let's find the values of the first few terms to understand the sequence better:
- \( f(0) = 108 \)
- \( f(1) = \frac{2}{3} \cdot 108 = 72 \)
- \( f(2) = \frac{2}{3} \cdot 72 = 48 \)
- \( f(3) = \frac{2}{3} \cdot 48 = 32 \) (approximately)
- \( f(4) = \frac{2}{3} \cdot 32 = 21.33 \) (approximately)
- \( f(5) = \frac{2}{3} \cdot 21.33 = 14.22 \) (approximately)
- We can continue finding more terms if needed, but let's stop here for clarity.
4. General Formula:
To derive a general formula for the sequence, observe the pattern:
- \( f(1) = \left(\frac{2}{3}\right) \cdot 108 \)
- \( f(2) = \left(\frac{2}{3}\right)^2 \cdot 108 \)
- \( f(3) = \left(\frac{2}{3}\right)^3 \cdot 108 \)
Generalizing this, we get:
[tex]\[
f(x) = \left(\frac{2}{3}\right)^x \cdot 108
\][/tex]
5. Graphing the Sequence:
To graph the sequence, plot the terms \( f(x) \) for integer values of \( x \).
Here are the first few points:
- \( (0, 108) \)
- \( (1, 72) \)
- \( (2, 48) \)
- \( (3, 32) \)
- \( (4, 21.33) \)
- \( (5, 14.22) \)
Plot these points on a graph where the x-axis represents the term number \( x \) and the y-axis represents the term value \( f(x) \).
6. Behavior of the Sequence:
- The sequence is decreasing and converges towards zero as \( x \) increases.
- It is a geometric sequence with a common ratio of \( \frac{2}{3} \).
Thus, the graph would show a series of points that start at [tex]\( (0, 108) \)[/tex] and rapidly decrease towards zero, illustrating the exponential decay described by [tex]\( f(x) = \left(\frac{2}{3}\right)^x \cdot 108 \)[/tex].
