Question

write a(subscript)n=3(subscript)1(9)^(n-1) into a geometric sequence

Answer 1

To express the formula you provided in standard notation for a geometric sequence, let's first clarify the notation.

The expression \( a_n = 3 \cdot 9^{(n-1)} \) represents the \( n \)-th term of a geometric sequence where:
- The first term (\( a_1 \)) occurs when \( n = 1 \).
- The common ratio (\( r \)) is derived from the base of the exponent, which is 9 in this case.

**Step 1: Calculate the first term \( a_1 \)**

Substituting \( n = 1 \):
\[
a_1 = 3 \cdot 9^{(1-1)} = 3 \cdot 9^0 = 3 \cdot 1 = 3
\]

**Step 2: Calculate the common ratio \( r \)**

The common ratio \( r \) can be determined by dividing the second term by the first term or recognizing it from the structure:
\[
r = 9 \quad \text{(as it's the base of the exponent)}
\]

**Step 3: Write the general form of the geometric sequence**

A geometric sequence can be represented generally as:
\[
a_n = a_1 \cdot r^{(n-1)}
\]
Substituting the known values:
- \( a_1 = 3 \)
- \( r = 9 \)

The sequence can be written as:
\[
a_n = 3 \cdot 9^{(n-1)}
\]

This confirms that the terms of your sequence are of the form \( 3, 27, 243, \ldots \) as you calculate \( a_2, a_3, \) etc. explicitly.

**Conclusion:**

Thus, \( 3 \cdot 9^{(n-1)} \) is already in the standard form of a geometric sequence where the first term is 3 and the common ratio is 9.