Answer :

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].