Correct answer: (a) 1/2

Explanation:

a + ar + ar² + .....∞ = 15

⟹ \(\frac{a}{1 - r}\) = 15  ....(1)

a² + (ar)² + (ar²)² + ....∞ = 150

⟹ \(\frac{a^2}{1 - r^2}\) = 150  ....(2)

Square (1) and divide by 2

\(\frac{\frac{a^2}{1-r^2}}{\frac{a^2}{(1-r)^2}}\) = \(\frac{150}{15.15}\)

⟹ \(\frac{1 - r}{1 + r}\) =  \(\frac{2}{3}\)

⟹ 3 - 3r = 2 + 2r

⟹ 1 = 5r

⟹ r = \(\frac{1}{5}\)

Substitute in (1)

\(\frac{a}{1 - \frac{1}{5}}\) = 15

\(\frac{5a}{4}\) = 15

a = 12

ar³ + ar⁴ + ar⁶ + ...∞

= \(\frac{ar^2}{1 + r^2}\)

= \(\frac{12 \times (\frac{1}{5})^2}{1 - (\frac{1}{5})^2}\)

= \(\frac{\frac{12}{25}}{\frac{24}{25}}\) = \(\frac{1}{2}\)