The second term of a geometric progression is 3. If its sum to infinity is

Explanation

\(T_{n} = ar^{n - 1}\) (Terms of a geometric progression)

\(T_{2} = ar = 3 ..... (1)\)

\(S_{\infty} = \frac{a}{1 - r}\) (Sum to infinity of a GP)

\(\frac{a}{1 - r} = \frac{25}{2} \implies 2a = 25(1 - r)\)

\(a = \frac{25 - 25r}{2}\)

\(3 = (\frac{25 - 25r}{2})r\)

\(6 = 25r - 25r^{2}\)

\(25r^{2} - 25r + 6 = 0\)

\(25r^{2} - 15r - 10r + 6 = 0\)

\(5r(5r - 3) - 2(5r - 3) = 0 \implies (5r - 3)(5r - 2) = 0\)

\(r = \frac{2}{5} \text{or r = } \frac{3}{5}\).