Question

what are the first three terms of a geometric sequence in which $a_5 = 25$ and the common ratio is 5?
$\frac{1}{25}, \frac{1}{5}, 1$
$25, 125, 625$
$\frac{1}{25}, \frac{1}{125}, \frac{1}{625}$
$125, 25, 5$

Explanation:

Step1: Recall geometric sequence formula

The formula for the \(n\)-th term of a geometric sequence is \(a_n = a_1r^{n - 1}\), where \(a_n\) is the \(n\)-th term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

Step2: Substitute known values

We know that \(a_5 = 25\), \(r = 5\), and \(n = 5\). Substitute these into the formula: \(25=a_1\times5^{5 - 1}\)

Step3: Simplify the equation

Simplify \(5^{5 - 1}=5^4 = 625\). So the equation becomes \(25=a_1\times625\)

Step4: Solve for \(a_1\)

To find \(a_1\), divide both sides of the equation by 625: \(a_1=\frac{25}{625}=\frac{1}{25}\)

Step5: Find the second term (\(a_2\))

Using the formula \(a_n = a_1r^{n - 1}\) with \(n = 2\), \(a_1=\frac{1}{25}\), and \(r = 5\): \(a_2=\frac{1}{25}\times5^{2 - 1}=\frac{1}{25}\times5=\frac{1}{5}\)

Step6: Find the third term (\(a_3\))

Using the formula with \(n = 3\), \(a_1=\frac{1}{25}\), and \(r = 5\): \(a_3=\frac{1}{25}\times5^{3 - 1}=\frac{1}{25}\times25 = 1\)

Answer:

\(\frac{1}{25},\frac{1}{5},1\) (the first option: \(\boldsymbol{\frac{1}{25},\frac{1}{5},1}\))