CAT 2024 Slot 2 QA Q8 Progressions & Series

Question 8 of the Quantitative Ability (QA) section, CAT 2024 Slot 2, with CAT's official answer and a worked solution.

The sum of the infinite series \(\frac{1}{5}\left(\frac{1}{5}-\frac{1}{7}\right)+\left(\frac{1}{5}\right)^2\left(\left(\frac{1}{5}\right)^2-\left(\frac{1}{7}\right)^2\right)+\left(\frac{1}{5}\right)^3\left(\left(\frac{1}{5}\right)^3-\left(\frac{1}{7}\right)^3\right)+\cdots\) is equal to

A
\(\frac{5}{408}\)
B
\(\frac{7}{816}\)
C
\(\frac{7}{408}\)
D
\(\frac{5}{816}\)

Answer: (A) 5/408 Progressions & Series MCQ

Solution

The \(n\)-th term is \(\left(\frac{1}{5}\right)^n\left(\frac{1}{5^n} - \frac{1}{7^n}\right) = \frac{1}{25^n} - \frac{1}{35^n}\).
Sum of two GPs: \(\sum \frac{1}{25^n} = \frac{1/25}{1-1/25} = \frac{1}{24}\), \(\sum \frac{1}{35^n} = \frac{1}{34}\).
\(\frac{1}{24} - \frac{1}{34} = \frac{34-24}{816} = \frac{10}{816} = \frac{5}{408}\).

Shortcut. For ratio \(r\), \(\sum_{n\ge1} r^n = \frac{r}{1-r}\); with \(r = 1/k\) this is \(\frac{1}{k-1}\).