Question

Consider the sequence 1,3,9,27,81...

What is the function that represents the sequence for the term numbers 1,2,3,4 and 5

Answer 1

The sequence you provided is 1, 3, 9, 27, 81. This can be recognized as a geometric sequence where each term is derived from multiplying the previous term by 3.

To express this in a mathematical function, we can represent the \( n \)-th term of the sequence with the function:

\[
a_n = 3^{(n-1)}
\]

Where:
- \( n \) is the term number (1 for the first term, 2 for the second term, etc.).
- \( a_n \) is the value of the \( n \)-th term.

Now let's verify the function for the first five terms:

1. For \( n = 1 \):
\[
a_1 = 3^{(1-1)} = 3^0 = 1
\]

2. For \( n = 2 \):
\[
a_2 = 3^{(2-1)} = 3^1 = 3
\]

3. For \( n = 3 \):
\[
a_3 = 3^{(3-1)} = 3^2 = 9
\]

4. For \( n = 4 \):
\[
a_4 = 3^{(4-1)} = 3^3 = 27
\]

5. For \( n = 5 \):
\[
a_5 = 3^{(5-1)} = 3^4 = 81
\]

Thus, the function \( a_n = 3^{(n-1)} \) correctly represents the terms of the sequence for \( n = 1, 2, 3, 4, \) and \( 5 \).