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.