Question

The 25th term of 9, 3, 1, \( \frac{1}{3} \), \( \frac{1}{9} \), ... is:

• A. \( \frac{1}{3^{24}} \)
• B. \( \frac{1}{3^{25}} \)
• C. \( \frac{1}{3^{23}} \)
• D. \( \frac{1}{3^{22}} \)
• E. \( \frac{1}{3^{26}} \)

Hint:

For a geometric sequence, the \(n\)-th term is calculated using the formula \(T_n = a \cdot r^{n-1}\), where \(a\) is the first term and \(r\) is the common ratio.

Answer 1

The Correct Option is D

Step 1:Understand the type of sequence.
The given sequence is:
\[ 9, 3, 1, \frac{1}{3}, \frac{1}{9}, \dots \] This is a geometric sequence where each term is obtained by multiplying the previous term by a constant ratio.

Step 2:Identify the first term and the common ratio.
- The first term \( a = 9 \).
- The common ratio \( r = \frac{3}{9} = \frac{1}{3} \).

Step 3:Formula for the \(n\)-th term of a geometric sequence.
The general formula for the \(n\)-th term of a geometric sequence is:
\[ T_n = a \cdot r^{n-1} \] where:
- \( a \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the term number.

Step 4:Substitute the values to find the 25th term.
We are asked to find the 25th term (\(T_{25}\)) of the sequence:
\[ T_{25} = 9 \cdot \left( \frac{1}{3} \right)^{25-1} = 9 \cdot \left( \frac{1}{3} \right)^{24} \] Simplifying:
\[ T_{25} = \frac{9}{3^{24}} = \frac{1}{3^{22}} \]
Step 5:Final Answer.
Therefore, the 25th term of the sequence is \( \frac{1}{3^{22}} \).