$d(n)=3(-2)^{n-1}$
What is the $4^{\text {th }}$ term in the sequence?
We are given the sequence formula \(d(n) = 3(-2)^{n-1}\), which defines the terms of a geometric sequence. In this formula, \(n\) represents the position of the term in the sequence. Our goal is to find the value of the \(4^{\text{th}}\) term, which means we need to evaluate the expression when \(n=4\). The first term of this sequence is \(d(1) = 3(-2)^{1-1} = 3(1) = 3\), and the common ratio is \(-2\).
To find the \(4^{\text{th}}\) term, we substitute \(n=4\) into the sequence formula \(d(n) = 3(-2)^{n-1}\). This gives us:
\[d(4) = 3(-2)^{4-1}\]
First, we simplify the exponent:
\[4 - 1 = 3\]
So, the expression becomes:
\[d(4) = 3(-2)^3\]
Now, we calculate the power \((-2)^3\). Since the exponent is odd, the negative sign remains:
\[(-2)^3 = (-2) \times (-2) \times (-2) = -8\]
Finally, we multiply this result by the coefficient 3 to obtain the \(4^{\text{th}}\) term:
\[d(4) = 3 \times (-8) = -24\]
Thus, the \(4^{\text{th}}\) term of the sequence is \(-24\).
This sequence formula is a practical tool for modeling processes that change by a constant multiplicative factor. For instance, in biology, a cell might divide into two identical daughter cells in each cycle, but if the process also involves a scaling factor (like a constant supply of nutrients or a mortality rate), the number of cells can be modeled by a geometric sequence like this one. Understanding how to calculate specific terms in such sequences allows scientists to predict population growth or decay over discrete intervals of time.