The second term of a geometric sequence is 6 and the 5th term is 48 determine the sequence

To determine the geometric sequence, we need to find the first term (\(a_1\)) and the common ratio (\(r\)) of the sequence. We know that:

- The second term \(a_2 = 6\)
- The fifth term \(a_5 = 48\)

The general formula for the \(n\)th term of a geometric sequence is:

\[
a_n = a_1 \cdot r^{(n-1)}
\]

Using this formula, we can express the second and fifth terms:

\[
a_2 = a_1 \cdot r^{(2-1)} = a_1 \cdot r = 6
\]

\[
a_5 = a_1 \cdot r^{(5-1)} = a_1 \cdot r^4 = 48
\]

Now, divide the equation for the fifth term by the equation for the second term:

\[
\frac{a_5}{a_2} = \frac{a_1 \cdot r^4}{a_1 \cdot r} = r^3 = \frac{48}{6} = 8
\]

Thus,

\[
r^3 = 8 \quad \Rightarrow \quad r = \sqrt[3]{8} = 2
\]

Now that we have \(r = 2\), substitute this back into the equation for the second term to find \(a_1\):

\[
a_1 \cdot 2 = 6 \quad \Rightarrow \quad a_1 = \frac{6}{2} = 3
\]

So, the first term is \(a_1 = 3\) and the common ratio is \(r = 2\). Therefore, the sequence is:

\[
3, 6, 12, 24, 48, \ldots
\]

This is the geometric sequence.

Would you like more details or have any questions about this? Here are some related questions:

1. What is the 10th term of this sequence?
2. How would the sequence change if the common ratio were -2?
3. What is the sum of the first 5 terms of this sequence?
4. How do you find the nth term of a geometric sequence?
5. What if the second term was given as 12 instead of 6?

**Tip:** The common ratio in a geometric sequence can be found by dividing any term by its previous term.

Learn how to determine the geometric sequence with given terms. Discover the first term and common ratio using the second term 6 and the fifth term 48. Detailed step-by-step solution provided.

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