In a geometric sequence, each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Let's look at the provided sequence:
- \( a_1 = 7 \)
- \( a_2 = 21 \)
- \( a_3 = 63 \)
- \( a_4 = 189 \)
First, we can find the common ratio \( r \):
- To go from \( 7 \) to \( 21 \): \( r = \frac{21}{7} = 3 \)
- To go from \( 21 \) to \( 63 \): \( r = \frac{63}{21} = 3 \)
- To go from \( 63 \) to \( 189 \): \( r = \frac{189}{63} = 3 \)
Thus, the common ratio \( r \) is \( 3 \).
In a geometric sequence, the \( n \)-th term can be expressed as: \[ a_n = a_1 \cdot r^{(n-1)} \]
Substituting in our values:
- \( a_1 = 7 \)
- \( r = 3 \)
So the formula for the \( n \)-th term becomes: \[ a_n = 7 \cdot 3^{(n-1)} \]
To express this in the form of an exponential equation \( y = ()x \), we can relate it to \( y \) and \( x \) by letting \( x = n \): \[ y = 7 \cdot 3^{(x-1)} \]
This can also be simplified: \[ y = \frac{7}{3} \cdot 3^x \]
If you want a simpler version focusing just on \( n \) as a variable, you can write: \[ y = 7 \cdot 3^{x-1} \]
Thus, the exponential equation for the geometric sequence is: \[ y = 7 \cdot 3^{(x-1)} \]
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