Question

Write the explicit formula for the geometric sequence below.
4​, 12​, 36​, 108​, ...
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Part 1
a Subscript nequals
  
enter your response heretimesleft parenthesis nothing right parenthesis Superscript n minus 1
  
​, for ngreater than or equals1
​(Simplify your answers. Type integers or​ fractions.

Answer 1

To write the explicit formula for the geometric sequence provided, we need to identify the first term and the common ratio of the sequence.

The sequence is: 4, 12, 36, 108, ...

1. **Identify the first term (a₁)**:
- The first term \( a_1 = 4 \).

2. **Identify the common ratio (r)**:
- To find the common ratio, divide any term by the previous term.
- Common ratio \( r = \frac{a_2}{a_1} = \frac{12}{4} = 3 \).
- We can verify this by checking a couple of terms:
- \( \frac{a_3}{a_2} = \frac{36}{12} = 3 \)
- \( \frac{a_4}{a_3} = \frac{108}{36} = 3 \)

So the common ratio \( r = 3 \).

3. **Write the explicit formula**:
The explicit formula for a geometric sequence is given by:
\[
a_n = a_1 \cdot r^{n-1}
\]

Substituting the known values into the formula:
\[
a_n = 4 \cdot 3^{n-1}
\]

Thus, the final explicit formula for the sequence is:
\[
\boxed{a_n = 4 \cdot 3^{n-1}} \quad \text{for } n \geq 1
\]