Question

Find an explicit formula for the geometric sequence
\[12\,,\,24\,,\,48\,,\,96,\unicode{0x2026}\].
Note: the first term should be
\[\text{\it{b(1)}}\].
\[b(n)=\]

Answer 1

To identify the explicit formula for the given geometric sequence \(12, 24, 48, 96, \ldots\), we need to determine the first term and the common ratio.

1. **First Term ( \(b(1)\) )**:
The first term \(b(1)\) is \(12\).

2. **Common Ratio (r)**:
We can find the common ratio by dividing the second term by the first term:
\[
r = \frac{24}{12} = 2
\]
To confirm, we check the ratio between other consecutive terms:
\[
\frac{48}{24} = 2 \quad \text{and} \quad \frac{96}{48} = 2
\]
Therefore, the common ratio is indeed \(2\).

3. **General Formula**:
In a geometric sequence, the \(n\)-th term can be expressed with the formula:
\[
b(n) = b(1) \cdot r^{n-1}
\]
Substituting the values we found:
\[
b(n) = 12 \cdot 2^{n-1}
\]

Thus, the explicit formula for the given geometric sequence is:
\[
\boxed{12 \cdot 2^{n-1}}
\]