in a geometric sequence a1=125 and a3=5 find the common ratio

1. The formula for the \(n\)-th term of a geometric sequence is given by:

a_n = a_1 \cdot r^{n-1}

2. Given \( a_1 = 125 \) and \( a_3 = 5 \). We can substitute into the formula for the third term:

a_3 = a_1 \cdot r^{3-1}

3. Substitute the known values:

5 = 125 \cdot r^2

4. Solve for \( r^2 \):

r^2 = \frac{5}{125}

5. Simplify the fraction:

r^2 = \frac{1}{25}

6. Take the square root of both sides to solve for \( r \):

r=\pm\frac{1}{5}

7. Therefore, the common ratio is:

r=\pm\frac{1}{5}