Question
Given the first five terms of the sequence 81, 27, 9, 3, 1, 1/3... use the subscript notation to describe a8
Answers
Answered by
GPT-4o mini
The sequence you've provided is a geometric sequence where each term is obtained by multiplying the previous term by \(\frac{1}{3}\). The first term \(a_1\) is 81, the second term \(a_2\) is 27, the third term \(a_3\) is 9, the fourth term \(a_4\) is 3, the fifth term \(a_5\) is 1, and the sixth term \(a_6\) is \(\frac{1}{3}\).
To write a general formula for the \(n\)-th term \(a_n\) of this geometric sequence, we can observe that:
- The first term \(a_1 = 81\) can be expressed as \(81 = 3^4\).
- Each subsequent term is a division by 3 (or a multiplication by \(\frac{1}{3}\)).
The \(n\)-th term can therefore be written as:
\[
a_n = 81 \cdot \left( \frac{1}{3} \right)^{n-1} = 3^4 \cdot \left( \frac{1}{3} \right)^{n-1} = 3^4 \cdot 3^{-(n-1)} = 3^{4 - (n-1)} = 3^{5 - n}
\]
Now, to find \(a_8\):
\[
a_8 = 3^{5 - 8} = 3^{-3} = \frac{1}{3^3} = \frac{1}{27}
\]
So, the value of \(a_8\) is \(\frac{1}{27}\).
To write a general formula for the \(n\)-th term \(a_n\) of this geometric sequence, we can observe that:
- The first term \(a_1 = 81\) can be expressed as \(81 = 3^4\).
- Each subsequent term is a division by 3 (or a multiplication by \(\frac{1}{3}\)).
The \(n\)-th term can therefore be written as:
\[
a_n = 81 \cdot \left( \frac{1}{3} \right)^{n-1} = 3^4 \cdot \left( \frac{1}{3} \right)^{n-1} = 3^4 \cdot 3^{-(n-1)} = 3^{4 - (n-1)} = 3^{5 - n}
\]
Now, to find \(a_8\):
\[
a_8 = 3^{5 - 8} = 3^{-3} = \frac{1}{3^3} = \frac{1}{27}
\]
So, the value of \(a_8\) is \(\frac{1}{27}\).