Infinite Geometric Progression — CAT Previous-Year Questions

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

The value of 1 + $\left(1+\frac{1}{3}\right)\frac{1}{4}$ + $\left(1+\frac{1}{3}+\frac{1}{9}\right)\frac{1}{16}$ + $\left(1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}\right)\frac{1}{64}$ + ..., is

An infinite geometric progression a₁, a₂, a₃, … has the property that a_n = 3(a_n+1 + a_n+2 + …) for every n ≥ 1. If the sum a₁ + a₂ + a₃ + … = 32, then a₅ is

Consider the sequence of numbers a₁, a₂, a₃, ... to infinity where a₁ = 81.33 and a₂ = –19 and a_j = a_j–1 – a_j–2 for j ≥ 3. What is the sum of the first 6002 terms of this sequence?

Each question is followed by two statements, A and B. Answer each question using the following instructions

Choose 1 if the question can be answered by using one of the statements alone but not by using the other statement alone.
Choose 2 if the question can be answered by using either of the statements alone.
Choose 3 if the question can be answered by using both statements together but not by either statement alone.
Choose 4 if the question cannot be answered on the basis of the two statements.

Is $\left(\frac{1}{a^2}+\frac{1}{a^4}+\frac{1}{a^6}+...\right)>\left(\frac{1}{a}+\frac{1}{a^3}+\frac{1}{a^5}+...\right)?$

A. −3 ≤ a ≤ 3
B. One of the roots of the equation 4x2 − 4x + 1 = 0 is a

The infinite sum $1+\frac{4}{7}+\frac{9}{7^2}+\frac{16}{7^3}+\frac{25}{7^4}+....$ equals