Progressions — CAT Previous-Year Questions

Suppose x₁, x₂, x₃, ..., x₁₀₀ are in arithmetic progression such that x₅ = -4 and 2x₆ + 2x₉ = x₁₁ + x₁₃. Then x₁₀₀ equals

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

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

For some positive and distinct real numbers x, y and z if $\frac{1}{\sqrt{\mathrm{y}}+\sqrt{\mathrm{z}}}$ is the arithmetic mean of $\frac{1}{\sqrt{\mathrm{x}}+\sqrt{\mathrm{z}}}$ and $\frac{1}{\sqrt{\mathrm{x}}+\sqrt{y}}$, then the relationship which will always hold true, is?

Let both the series a₁, a₂, a₃, ... and b₁, b₂, b₃, ... be in arithmetic progression such that the common differences of both the series are prime numbers. If a₅ = b₉, a₁₉ = b₁₉ and b₂ = 0, then a₁₁ equal?

Let a_n and b_n be two sequences such that a_n = 13 + 6(n - 1) and b_n = 15 + 7(n - 1) for all natural numbers n. Then, the largest three digit integer that is common to both these sequences is

Let a_n = 46 + 8n and b_n = 98 + 4n be two sequences for natural numbers n ≤ 100. Then, the sum of all terms common to both the sequences is

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

For any natural number n, suppose the sum of the first n terms of an arithmetic progression is (n+ 2n²). If the n^th term of the progression is divisible by 9, then the smallest possible value of n is

Consider the arithmetic progressions 3, 7, 11, ... and let An dentoe the sum of the first n terms of this progression. Then the value of $\frac{1}{25}\underset{\mathrm{n}=1}{\overset{25}{\sum {\mathrm{A}}_{\mathrm{n}}}}$

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.

The average of all 3-digit terms in the arithmetic progression 38, 55, 72, ..., is

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.

Three positive integers x, y and z are in arithmetic progression. If y − x > 2 and xyz = 5(x + y + z), then z − x equals 

Consider a sequence of real numbers x₁, x₂, x₃, … such that x_n+1 = x_n + n – 1 for all n ≥ 1. If x₁ = -1 then x₁₀₀

Let the m-th and n-th terms of a geometric progression be 3/4 and 12, respectively, where m < n. If the common ratio of the progression is an integer r, then the smallest possible value of r + n - m is

If x₁ = - 1 and x_m = x_m+1 + (m + 1) for every positive integer m, then x₁₀₀ equals

If a₁ + a₂ + a₃ + … + a_n = 3(2ⁿ⁺¹ – 2), for every n ≥ 1, then a₁₁ equals 

If the population of a town is p in the beginning of any year then it becomes 3 + 2p in the beginning of the next year. If the population in the beginning of 2019 is 1000, then the population in the beginning of 2034 will be

If a₁, a₂, ... are in A.P., then, $\frac{1}{\sqrt{a_1}+\sqrt{a_2}}$ + $\frac{1}{\sqrt{a_2}+\sqrt{a_3}}$ + ... + $\frac{1}{\sqrt{a_n}+\sqrt{a_{n+1}}}$ is equal to

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

The number of common terms in the two sequences: 15, 19, 23, 27, ...... , 415 and 14, 19, 24, 29, ...... , 464 is

If (2n+1) + (2n+3) + (2n+5) + ... + (2n+47) = 5280 , then what is the value of 1 + 2 + 3 + ... + n?

Let x, y, z be three positive real numbers in a geometric progression such that x < y < z. If 5x, 16y, and 12z are in an arithmetic progression then the common ratio of the geometric progression is

The arithmetic mean of x, y and z is 80, and that of x, y, z, u and v is 75, where u = (x + y)/2 and v = (y + z)/2. If x ≥ z, then the minimum possible value of x is

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

Suppose, log₃x = log₁₂y = a, where x, y are positive numbers. If G is the geometric mean of x and y, and log₆G is equal to:

If the square of the 7^th term of an arithmetic progression with positive common difference equals the product of the 3^rd and 17^th terms, then the ratio of the first term to the common difference is:

Let a₁, a₂,.......a_3n be an arithmetic progression with a₁ = 3 and a₂ = 7. If a₁ + a₂ + ......+ a_3n = 1830, then what is the smallest positive integer m such that m(a₁ + a₂ + ..... + a_n) > 1830?

If log (2^a × 3^b × 5^c) is the arithmetic mean of log (2² × 3³ × 5), log (2⁶ × 3 × 5⁷), and log (2 × 3² × 5⁴), then a equals

Let a₁, a₂, a₃, a₄, a₅ be a sequence of five consecutive odd numbers. Consider a new sequence of five consecutive even numbers ending with 2a₃.

If the sum of the numbers in the new sequence is 450, then a₅ 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

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 number of common terms in the two sequences 17, 21, 25, … , 417 and 16, 21, 26, … , 466  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 $\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+.....+\sqrt{1+\frac{1}{{2007}^2}+\frac{1}{{2008}^2}}$

The price of Darjeeling tea (in rupees per kilogram) is 100 + 0.10n, on the n^th day of 2007 (n = 1, 2, ..., 100), and then remains constant. On the other hand, the price of Ooty tea (in rupees per kilogram) is 89 + 0.15n, on the n^th day of 2007 (n = 1, 2, ..., 365). On which date in 2007 will the prices of these two varieties of tea be equal?

Which of the following best describes $a_n+b_n$ for even n?

If p = 1/3 and q = 2/3, then what is the smallest odd n such that a_n + b_n < 0.01?

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:

A group of 630 children is arranged in rows for a group photograph session. Each row contains three fewer children than the row in front of it. What number of rows is not possible?

Consider the set S = {1, 2, 3, …, 1000}. How many arithmetic progressions can be formed from the elements of S that start with 1 and end with 1000 and have at least 3 elements?

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

If the sum of the first 11 terms of an arithmetic progression equals that of the first 19 terms, then what is the sum of the first 30 terms?

On January 1, 2004 two new societies, S₁, and S₂, are formed, each with n members. On the first day of each subsequent month, S1 adds b members while S2 multiplies its current number of members by a constant factor r. Both the societies have the same number of members on July 2, 2004. If b = 10.5n, what is the value of r?

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?

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?

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 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

If the product of n positive real numbers is unity, then their sum is necessarily

If log₃ 2, log₃ (2^x − 5), log₃ (2^x − 7/2) are in arithmetic progression, then the value of x is equal to

In a certain examination paper, there are n questions. For j = 1, 2, ..., n, there are 2^{(n − j)} students who answered j or more questions wrongly. If the total number of wrong answers is 4095, then the value of n is

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

On a straight road XY, 100 metres in length, 5 stones are kept beginning from the end X. The distance between two adjacent stones is 2 metres. A man is asked to collect the stones one at a time and put at the end Y. What is the distance covered by him?

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

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

A student finds the sum 1 + 2 + 3 + ... as his patience runs out. He found the sum as 575. When the teacher declared the result wrong, the student realized that he missed a number. What was the number the student missed?

Two men X and Y started working for a certain company at similar jobs on January 1, 1950. X asked for an initial salary of Rs. 300 with an annual increment of Rs. 30. Y asked for an initial salary of Rs. 200 with a rise of Rs. 15 every six months. Assume that the arrangements remained unaltered till December, 1959. Salary is paid on the last day of the month. What is the total amount paid to them as salary during the period?

All the page numbers from a book are added, beginning at page 1. However, one page number was mistakenly added twice. The sum obtained was 1000. Which page number was added twice?

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?

If a₁ = 1 and a_n+1 = 2a_n + 5, n = 1, 2 ... , then a₁₀₀ is equal to

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

First term of S₁ is

Second term of S₂ is

What is the difference between second and fourth terms of S₁?

What is the average value of the terms of series S₁?

What is the sum of series S₂?

If the harmonic mean between two positive numbers is to their geometric mean as 12 : 13; then the numbers could be in the ratio

Fourth term of an arithmetic progression is 8. What is the sum of the first 7 terms of the arithmetic progression?

Along a road lie an odd number of stones placed at intervals of 10m. These stones have to be assembled around the middle stone. A person can carry only one stone at a time. A man carried out the job starting with the stone in the middle, carrying stones in succession, thereby covering a distance of 4.8 km. Then the number of stones is

Let U_n+1 = 2U_n + 1 (n = 0, 1, 2, ...) and u₀ = 0. Then u₁₀ is nearest to

Let x < 0.50, 0 < y < 1, z > 1. Given a set of numbers, the middle number, when they are arranged in ascending order, is called the median. So the median of the numbers x, y, and z would be