Miscellaneous Progressions — CAT Previous-Year Questions

Consider the sequence t₁ = 1, t₂ = -1 and t_n = $\left(\frac{n-3}{n-1}\right)$t_n-2 for n ≥ 3. Then, the value of the sum $\frac{1}{t_2}$ + $\frac{1}{t_4}$ + $\frac{1}{t_6}$ + ... + $\frac{1}{t_{2022}}$ + $\frac{1}{t_{2024}}$

A lab experiment measures the number of organisms at 8 am every day. Starting with 2 organisms on the first day, the number of organisms on any day is equal to 3 more than twice the number on the previous day. If the number of organisms on the n^th day exceeds one million, then the lowest possible value of n is

On day one, there are 100 particles in a laboratory experiment. On day n, where n greater than or 2, one out of every n particles produces another particle. If the total number of particles in the laboratory experiment increases to 1000 on day m, then m equals.

If x₀ = 1, x₁ = 2 and x_n+2 = $\frac{1+x_{n+1}}{x_n}$, n = 0, 1, 2, 3, …, then x₂₀₂₁ is equal to

The natural numbers are divided into groups as (1), (2, 3, 4), (5, 6, 7, 8, 9), ….. and so on. Then, the sum of the numbers in the 15^th group is equal to

For a sequence of real numbers x₁, x₂, …, x_n, if x₁ - x₂ + x₃ - … + (-1)⁽ⁿ⁺¹⁾ x_n = n² + 2n for all natural numbers n, then the sum x₄₉ + x₅₀ equals.

Let a₁ , a₂ be integers such that a₁ - a₂ + a₃ - a₄ + ........ + (-1)^n-1 a_n = n , for n ≥ 1. Then a₅₁ + a₅₂ + ........ + a₁₀₂₃ equals

Let t₁, t₂,… be real numbers such that t₁ + t₂ + … + t_n = 2n² + 9n + 13, for every positive integer n ≥ 2. If t_k = 103, then k equals

The value of the sum 7 × 11 + 11 × 15 + 15 × 19 + ...+ 95 × 99 is

If a₁ = $\frac{1}{2\times 5}$, a₂ = $\frac{1}{5\times 8}$, a₃ = $\frac{1}{8\times 11}$, ......, then a₁ + a₂ + a₃ + .... + a₁₀₀ is

The integers 1, 2, …, 40 are written on a blackboard. The following operation is then repeated 39 times: In each repetition, any two numbers, say a and b, currently on the blackboard are erased and a new number a + b – 1 is written. What will be the number left on the board at the end?

Find the sum $\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+.....+\sqrt{1+\frac{1}{{2007}^2}+\frac{1}{{2008}^2}}$

Consider a sequence where the n^th term, $t_n=\frac{n}{n+2},n=1,2,....$

The value of $t_3\times t_4\times t_5\times ...\times t_{53}$ equals:

If a₁ = 1 and a_{n + 1} – 3a_n + 2 = 4n for every positive integer n, then a₁₀₀ equals

The sum of 3rd and 15th elements of an arithmetic progression is equal to the sum of 6th, 11th and 13th elements of the same progression. Then which element of the series should necessarily be equal to zero?

The 288th term of the series a, b, b, c, c, c, d, d, d, d, e, e, e, e, e, f, f, f, f, f, f... is

Let S = 2x + 5x² + 9x³ + 14x⁴ + 20x⁵ ... infinity (x < 1)

The coefficient of nth term = $\frac{n(n+3)}{2}.$ The sum is

For a Fibonacci sequence, from the third term onwards, each term in the sequence is the sum of the previous two terms in that sequence. If the difference in squares of seventh and sixth terms of this sequence is 517, what is the tenth term of this sequence?

What is the value of the following expression?

$\frac{1}{2^2-1}$ + $\frac{1}{4^2-1}$ + $\frac{1}{6^2-1}$ + ... + $\frac{1}{{20}^2-1}$

All values in S1 are changed in sign, while those in S2 remain unchanged. Which of the following statements is true?

Elements of S1 are in ascending order, and those of S2 are in descending order. a₂₄ and a₂₅ are interchanged. Then which of the following statements is true?

Every element of S1 is made greater than or equal to every element of S2 by adding to each  element of S1 an integer x. Then x cannot be less than