A sequence has a common ratio of Three-halves and f(5) = 81. Which explicit formula represents the sequence? f(x) = 24(Three-halves) Superscript x minus 1 f(x) = 16(Three-halves) Superscript x minus 1 f(x) = 24(Three-halves) Superscript x f(x) = 16(Three-halves) Superscript x
The Correct Answer and Explanation is:
The given problem describes a geometric sequence with a common ratio of ( \frac{3}{2} ) and ( f(5) = 81 ). To find the explicit formula for the sequence, we must first recall the general form of the explicit formula for a geometric sequence, which is:
[
f(x) = a \cdot r^{x – 1}
]
Where:
- ( f(x) ) is the value of the sequence at the position ( x ),
- ( a ) is the first term of the sequence,
- ( r ) is the common ratio, and
- ( x ) is the term number.
Step 1: Identify the common ratio and the form of the sequence
The problem states that the common ratio ( r ) is ( \frac{3}{2} ). This means that each term is multiplied by ( \frac{3}{2} ) to obtain the next term.
Step 2: Use the given value of ( f(5) )
We are also told that ( f(5) = 81 ). To find the value of the first term ( a ), we can substitute ( f(5) = 81 ) into the general formula:
[
f(5) = a \cdot \left( \frac{3}{2} \right)^{5 – 1}
]
This simplifies to:
[
81 = a \cdot \left( \frac{3}{2} \right)^4
]
Now, calculate ( \left( \frac{3}{2} \right)^4 ):
[
\left( \frac{3}{2} \right)^4 = \frac{3^4}{2^4} = \frac{81}{16}
]
So the equation becomes:
[
81 = a \cdot \frac{81}{16}
]
Step 3: Solve for ( a )
To solve for ( a ), multiply both sides of the equation by ( \frac{16}{81} ):
[
a = 81 \cdot \frac{16}{81} = 16
]
Thus, the first term of the sequence is ( a = 16 ).
Step 4: Write the explicit formula
Now that we know the first term ( a = 16 ) and the common ratio ( r = \frac{3}{2} ), we can substitute these values into the general formula:
[
f(x) = 16 \cdot \left( \frac{3}{2} \right)^{x – 1}
]
Conclusion
The explicit formula for the sequence is:
[
f(x) = 16 \cdot \left( \frac{3}{2} \right)^{x – 1}
]
This matches the second option in the problem. Therefore, the correct answer is:
[
f(x) = 16 \cdot \left( \frac{3}{2} \right)^{x – 1}
]