Which formula can be used to describe the sequence? $-\frac{2}{3},-4,-24,-144 \ldots$
A. $f(x)=6\left(-\frac{2}{3}\right)^{x-1}$
B. $f(x)=-6 \frac{2}{3}$
C. $f(x)=-\frac{2}{3}(6)^{x-1}$
D. $f(x)=\frac{2}{3} f-9,4$
Answer (1)
The sequence is geometric with a common ratio of 6.
Test the given formulas with x = 1 , 2 , 3 , 4 to see if they generate the sequence.
f ( x ) = 6 ( − 3 2 ) x − 1 does not generate the sequence.
f ( x ) = − 3 2 ( 6 ) x − 1 generates the sequence, so the final answer is f ( x ) = − 3 2 ( 6 ) x − 1 .
Explanation
Analyzing the Sequence We are given the sequence − 3 2 , − 4 , − 24 , − 144 , … and asked to find a formula that describes it from the given options.
First, let's check if the sequence is geometric. To do this, we will divide each term by the previous term to see if there is a common ratio.
Finding the Common Ratio − 3 2 − 4 = − 4 × 2 − 3 = 6 − 4 − 24 = 6 − 24 − 144 = 6
Since the ratio between consecutive terms is constant and equal to 6, the sequence is geometric with a common ratio of r = 6 .
Testing the Formulas Now, let's test the given formulas to see which one generates the sequence.
Option 1: f ( x ) = 6 ( − 3 2 ) x − 1 f ( 1 ) = 6 ( − 3 2 ) 1 − 1 = 6 ( − 3 2 ) 0 = 6 ( 1 ) = 6 . This does not match the first term of the sequence, which is − 3 2 .
Option 2: f ( x ) = − 3 2 ( 6 ) x − 1 f ( 1 ) = − 3 2 ( 6 ) 1 − 1 = − 3 2 ( 6 ) 0 = − 3 2 ( 1 ) = − 3 2 f ( 2 ) = − 3 2 ( 6 ) 2 − 1 = − 3 2 ( 6 ) 1 = − 3 2 ( 6 ) = − 4 f ( 3 ) = − 3 2 ( 6 ) 3 − 1 = − 3 2 ( 6 ) 2 = − 3 2 ( 36 ) = − 24 f ( 4 ) = − 3 2 ( 6 ) 4 − 1 = − 3 2 ( 6 ) 3 = − 3 2 ( 216 ) = − 144
This formula generates the correct sequence.
Option 3: f ( x ) = 3 2 ( − 9.4 ) x − 1 f ( 1 ) = 3 2 ( − 9.4 ) 1 − 1 = 3 2 ( − 9.4 ) 0 = 3 2 ( 1 ) = 3 2 . This does not match the first term of the sequence, which is − 3 2 .
Final Answer Therefore, the formula that describes the sequence is f ( x ) = − 3 2 ( 6 ) x − 1 .
Examples
Geometric sequences are useful in many real-world applications, such as calculating compound interest, modeling population growth, and determining the depreciation of assets. For example, if you invest $1000 in an account that earns 5% interest compounded annually, the amount of money you have each year forms a geometric sequence. Understanding geometric sequences helps you predict future values and make informed financial decisions.
