Arithmetic Progression — CAT Previous-Year Questions with Solutions

CAT 2024 Slot 1 · QA

Q1.

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

CAT 2023 Slot 2 · QA

Q2.

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?

CAT 2023 Slot 2 · QA

Q3.

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

CAT 2023 Slot 3 · QA

Q4.

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

CAT 2022 Slot 1 · QA

Q5.

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

CAT 2022 Slot 2 · QA

Q6.

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 $\frac{1}{25} {\sum A_{n}}_{n = 1}^{25}$

CAT 2022 Slot 3 · QA

Q7.

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

CAT 2021 Slot 2 · QA

Q8.

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

CAT 2021 Slot 3 · QA

Q9.

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

CAT 2020 Slot 3 · QA

Q10.

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

CAT 2019 Slot 1 · QA

Q11.

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

CAT 2019 Slot 2 · QA

Q12.

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

CAT 2019 Slot 2 · QA

Q13.

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

CAT 2017 Slot 1 · QA

Q14.

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:

CAT 2017 Slot 1 · QA

Q15.

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?

CAT 2017 Slot 2 · QA

Q16.

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

CAT 2008 · QA

Passage / Data

Answer the next 2 questions based on the information given below.

Let f(x) = ax² + bx + c, where, a, b and c are certain constants and a ≠ 0. It is known that f(5) = −3f(2) and that 3 is a root of f(x) = 0.

Q17.

The number of common terms in the two sequences 17, 21, 25, … , 417 and 16, 21, 26, … , 466 is

CAT 2007 · QA

Q18.

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?

CAT 2006 · QA

Q19.

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?

CAT 2006 · QA

Q20.

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?

CAT 2004 · QA

Q21.

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?

CAT 2003 Slot 1 · QA

Passage / Data

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.

Q22.

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

CAT 2002 · QA

Q23.

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?

CAT 2002 · QA

Passage / Data

Sum of first n natural numbers = S(n)

Sum given by student = 575

S(10) = $\frac{10 \times 11}{2} =$ 55

S(20) = $\frac{20 \times 21}{2} =$ 210

S(30) = $\frac{30 \times 31}{2} =$ 465

S(40) = $\frac{40 \times 41}{2} =$ 820

∴ The student stopped counting somewhere between 30 and 40.

Consider S(35) = $\frac{36 \times 35}{2} =$ 630

The student stopped somewhere before 35.

∴ S(31) = 496, S(32) = 528, S(33) = 561 and S(34) = 595

But the student gave 575 as the sum, so the student missed on the number 20.

Hence, option 4.

Q24.

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?

CAT 2001 · QA

Q25.

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?

CAT 2001 · QA

Q26.

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?

CAT 1996 · QA

Passage / Data