To determine the number of terms in the given geometric series [tex]\( 4, 8, 16, \ldots, 1024 \)[/tex], we can follow these steps:
1. Identify the initial term ([tex]\(a\)[/tex]) and the common ratio ([tex]\(r\)[/tex]) of the geometric series:
- The initial term ([tex]\(a\)[/tex]) is the first term of the series. Here, the first term is [tex]\(4\)[/tex].
- The common ratio ([tex]\(r\)[/tex]) is the ratio between any two consecutive terms in the series. For this series, we can find [tex]\(r\)[/tex] by dividing the second term by the first term:
[tex]\[
r = \frac{8}{4} = 2
\][/tex]
2. Recall the formula for the [tex]\(n\)[/tex]-th term of a geometric sequence:
The formula to find the [tex]\(n\)[/tex]-th term ([tex]\(T_n\)[/tex]) of a geometric sequence is given by:
[tex]\[
T_n = a \cdot r^{(n-1)}
\][/tex]
where [tex]\(a\)[/tex] is the initial term, [tex]\(r\)[/tex] is the common ratio, and [tex]\(n\)[/tex] is the number of terms.
3. Set up the equation for the last term of the series:
We need to find [tex]\(n\)[/tex] such that the [tex]\(n\)[/tex]-th term [tex]\(T_n\)[/tex] equals the last term given in the series, which is 1024. Thus, we have:
[tex]\[
1024 = 4 \cdot 2^{(n-1)}
\][/tex]
4. Solve for [tex]\(n\)[/tex]:
First, divide both sides of the equation by 4:
[tex]\[
\frac{1024}{4} = 2^{(n-1)}
\][/tex]
[tex]\[
256 = 2^{(n-1)}
\][/tex]
Next, express 256 as a power of 2:
[tex]\[
256 = 2^8
\][/tex]
Now we have:
[tex]\[
2^8 = 2^{(n-1)}
\][/tex]
Since the bases are the same, the exponents must be equal:
[tex]\[
n - 1 = 8
\][/tex]
Solving for [tex]\(n\)[/tex]:
[tex]\[
n = 8 + 1 = 9
\][/tex]
Therefore, the number of terms in the given geometric series is [tex]\(9\)[/tex].
