Asked by Gupgupblupblupzorpzorpflapflap
Find the exponential equation for the sequence: (0,3),(1,3.6),(2,4.32),(3,5.184)…
y = ?(?)^x
y = ?(?)^x
Answers
Answered by
Gupgupblupblupzorpzorpflapflap
Find the common ratio for the sequence: (0,3),(1,3.6),(2,4.32),(3,5.184)…
Answered by
GPT-4o mini
To find the exponential equation for the given sequence \((0,3),(1,3.6),(2,4.32),(3,5.184)...\), we can assume the general form of the exponential equation is:
\[
y = a \cdot b^x
\]
where \(a\) represents the initial value (the y-value when \(x=0\)) and \(b\) is the growth factor.
1. **Find the value of \(a\):**
From the point \((0, 3)\):
\[
y(0) = a \cdot b^0 = a = 3
\]
So we have:
\[
a = 3
\]
2. **Find the growth factor \(b\):**
Next, we can use the other points to determine \(b\). We can use the point \((1, 3.6)\):
\[
y(1) = 3 \cdot b^1 = 3 \cdot b = 3.6
\]
To solve for \(b\):
\[
b = \frac{3.6}{3} = 1.2
\]
3. **Verification of \(b\):**
To ensure that this value of \(b\) fits the other data points, we can calculate the next points using \(b = 1.2\).
For \(x=2\):
\[
y(2) = 3 \cdot (1.2)^2 = 3 \cdot 1.44 = 4.32
\]
For \(x=3\):
\[
y(3) = 3 \cdot (1.2)^3 = 3 \cdot 1.728 = 5.184
\]
Both match the given points in the sequence.
4. **Final equation:**
Thus, the exponential equation for the sequence is:
\[
y = 3(1.2)^x
\]
\[
y = a \cdot b^x
\]
where \(a\) represents the initial value (the y-value when \(x=0\)) and \(b\) is the growth factor.
1. **Find the value of \(a\):**
From the point \((0, 3)\):
\[
y(0) = a \cdot b^0 = a = 3
\]
So we have:
\[
a = 3
\]
2. **Find the growth factor \(b\):**
Next, we can use the other points to determine \(b\). We can use the point \((1, 3.6)\):
\[
y(1) = 3 \cdot b^1 = 3 \cdot b = 3.6
\]
To solve for \(b\):
\[
b = \frac{3.6}{3} = 1.2
\]
3. **Verification of \(b\):**
To ensure that this value of \(b\) fits the other data points, we can calculate the next points using \(b = 1.2\).
For \(x=2\):
\[
y(2) = 3 \cdot (1.2)^2 = 3 \cdot 1.44 = 4.32
\]
For \(x=3\):
\[
y(3) = 3 \cdot (1.2)^3 = 3 \cdot 1.728 = 5.184
\]
Both match the given points in the sequence.
4. **Final equation:**
Thus, the exponential equation for the sequence is:
\[
y = 3(1.2)^x
\]
Answered by
GPT-4o mini
To find the common ratio of the sequence given by the points \((0,3),(1,3.6),(2,4.32),(3,5.184)...\), we first note that this is not a geometric sequence but rather follows the form of an exponential function.
However, we can still find the ratio of successive terms to understand the growth factor of the sequence:
1. **Calculate the first common ratio** between the first and second terms:
\[
r_1 = \frac{y(1)}{y(0)} = \frac{3.6}{3} = 1.2
\]
2. **Calculate the second common ratio** between the second and third terms:
\[
r_2 = \frac{y(2)}{y(1)} = \frac{4.32}{3.6} = 1.2
\]
3. **Calculate the third common ratio** between the third and fourth terms:
\[
r_3 = \frac{y(3)}{y(2)} = \frac{5.184}{4.32} = 1.2
\]
Since each of these ratios is equal to \(1.2\), we can conclude that the common ratio for this sequence—reflecting the growth factor of the exponential function—is:
\[
\text{Common ratio} = 1.2
\]
However, we can still find the ratio of successive terms to understand the growth factor of the sequence:
1. **Calculate the first common ratio** between the first and second terms:
\[
r_1 = \frac{y(1)}{y(0)} = \frac{3.6}{3} = 1.2
\]
2. **Calculate the second common ratio** between the second and third terms:
\[
r_2 = \frac{y(2)}{y(1)} = \frac{4.32}{3.6} = 1.2
\]
3. **Calculate the third common ratio** between the third and fourth terms:
\[
r_3 = \frac{y(3)}{y(2)} = \frac{5.184}{4.32} = 1.2
\]
Since each of these ratios is equal to \(1.2\), we can conclude that the common ratio for this sequence—reflecting the growth factor of the exponential function—is:
\[
\text{Common ratio} = 1.2
\]