A sequence has a common ratio of 3/2 and f(5)=81. Which explicit formula represents the sequence?f(x)=24( 3/2 )^(x-1)f(x)=16( 3/2 )^(x-1)f(x)=24( 3/2 )^xf(x)=16( 3/2 )^x

Given the sequence has a common ratio of \(\frac { 3} { 2} \) and \(f( 5) = 81\), we need to find the explicit formula. The general form of a geometric sequence is \(f( x) = a \cdot r^ { x- 1} \), where \(a\) is the first term and \(r\) is the common ratio.

1. Let's denote the first term by \(a\).
2. We know \(f( 5) = 81\), so:
   \[a \cdot \left ( \frac { 3} { 2} \right ) ^ { 5- 1} = 81\]
   \[a \cdot \left ( \frac { 3} { 2} \right ) ^ 4 = 81\]
3. Calculate \(\left ( \frac { 3} { 2} \right ) ^ 4\):
   \[\left ( \frac { 3} { 2} \right ) ^ 4 = \frac { 81} { 16} \]
4. Substitute back into the equation:
   \[a \cdot \frac { 81} { 16} = 81\]
5. Solve for \(a\):
   \[a = 81 \cdot \frac { 16} { 81} = 16\]

Therefore, the explicit formula is:
\[f( x) = 16 \left ( \frac { 3} { 2} \right ) ^ { x- 1} \]

Supplemental Knowledge

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The explicit formula for a geometric sequence can be written as:
\[f( x) = a \cdot r^ { ( x- 1) } \]
where:

• \(a\) is the first term of the sequence,
• \(r\) is the common ratio,
• \(x\) is the position of the term in the sequence.

Given that \(f( 5) = 81\) and the common ratio \(r = 3/2\), we can use this information to determine the explicit formula.

Everyday Examples

Imagine yourself as an investor closely tracking their investments over time. If they grow by an agreed upon percentage every year, a geometric sequence model would help. For instance, if they double in value each year (a common ratio), and you know its current value at any particular moment in time; you could predict its future value using an explicit formula similar to geometric sequences.
This concept allows investors to understand how their investments will grow over time and make informed decisions regarding their financial strategies.

Geometric sequences offer valuable insight into a host of real-world phenomena ranging from financial investments to biological development patterns. If you want to master and apply these concepts effectively, UpStudy provides exceptional resources!

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