The second term of a geometric progression is…

Question 1 Report

The second term of a geometric progression is 3. If its sum to infinity is \(\frac{25}{2}\), find the value of its common ratio.

Answer

Answer Details

Let the first term be \(a\) and common ratio \(r\), with \(|r| < 1\).

Second term: \(ar = 3\), so \(a = \dfrac{3}{r}\).

Sum to infinity: \(\dfrac{a}{1 - r} = \dfrac{25}{2}\). Substitute \(a\):

\[\frac{3/r}{1 - r} = \frac{25}{2} \ \Rightarrow\ \frac{3}{r(1 - r)} = \frac{25}{2}\] \[6 = 25r(1 - r) \ \Rightarrow\ 25r^2 - 25r + 6 = 0\]

Solve:

\[r = \frac{25 \pm \sqrt{625 - 600}}{50} = \frac{25 \pm 5}{50}\] \[r = \frac{3}{5} \quad\text{or}\quad r = \frac{2}{5}\]

Both satisfy \(|r| < 1\), so the common ratio is \(r = \dfrac{3}{5}\) or \(r = \dfrac{2}{5}\).

The second term of a geometric progression is 3. If its sum to infinity is \(\frac{25}{2}\), find the value of its common ratio.