Question

write a formula for the nth term of the following geometric
3, 12, 48, ...
enter a formula for the nth term of the geometric sequence.
a_n = \square \cdot (\square)^{n - 1}

Explanation:

Step1: Identify the first term ($a_1$)

In a geometric sequence, the first term \( a_1 \) is the initial number. For the sequence \( 3, 12, 48, \dots \), the first term \( a_1 = 3 \).

Step2: Find the common ratio ($r$)

The common ratio \( r \) in a geometric sequence is found by dividing any term by the previous term. So, \( r=\frac{12}{3} = 4 \) (or \( \frac{48}{12}=4 \)).

Step3: Recall the formula for the nth term of a geometric sequence

The formula for the nth term of a geometric sequence is \( a_n=a_1\cdot r^{n - 1} \). Substituting \( a_1 = 3 \) and \( r = 4 \) into the formula, we get \( a_n=3\cdot(4)^{n - 1} \).

Answer:

\( a_n = \boldsymbol{3} \cdot (\boldsymbol{4})^{n - 1} \)