Question: What are the first three terms of a geometric sequence in which \( a_5 = 25 \) and the common…

What are the first three terms of a geometric sequence in which \( a_5 = 25 \) and the common ratio is 5?

1. \( 125, 25, 5 \)
2. \( \frac{1}{25}, \frac{1}{5}, 1 \)
3. \( 25, 125, 625 \)
4. \( \frac{1}{25}, \frac{1}{125}, \frac{1}{625} \)

Solution

The problem asks for the first three terms of a geometric sequence where \(a_5 = 25\) and the common ratio is 5. 1. The general formula for the \(n\)-th term of a geometric sequence is given by: \[ a_n = a_1 \cdot r^{n-1} \] 2. Substitute the given values for \(a_5\) and \(r\): \[ a_5 = a_1 \cdot 5^{4} = 25 \] 3. Solve for \(a_1\): \[ a_1 = \frac{25}{5^4} \] \[ a_1 = \frac{25}{625} \] \[ a_1 = \frac{1}{25} \] 4. Now, find the first three terms using \(a_1\) and the common ratio 5: \[ a_1 = \frac{1}{25} \] \[ a_2 = a_1 \cdot 5 = \frac{1}{25} \cdot 5 = \frac{1}{5} \] \[ a_3 = a_2 \cdot 5 = \frac{1}{5} \cdot 5 = 1 \] Thus, the first three terms of the sequence are \( \frac{1}{25}, \frac{1}{5}, 1 \).