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CAT Questions | CAT Algebra Questions

CAT Quantitative Aptitude | CAT Algebra: Progressions Questions

A CAT Algebra question from Progressions that appears in the Quantitative Aptitude section of the CAT Exam consists of concepts from Number Theory and Algebra. It involves concepts based on Artihmetic Progressions and Geometric Progressions. With some simple but very powerful ideas, one can cut down on a lot of working when it comes to progressions. For example, anchoring a progression around its middle term can be very useful. Reinforce these ideas with the following questions. In CAT Exam, one can generally expect to get 1~2 questions from Progressions. Make use of 2IIMs Free CAT Questions, provided with detailed solutions and Video explanations to obtain a wonderful CAT score. If you would like to take these questions as a Quiz, head on here to take these questions in a test format, absolutely free.

CAT Progressions - Geometric Progressions

Second term of a GP is 1000 and the common ratio is r = \\frac{1}{n}\\) where n is a natural number. P_n is the product of n terms of this GP. P₆ > P₅ and P₆ > P₇, what is the sum of all possible values of n?
1. 4
2. 9
3. 5
4. 13
Choice B
9
CAT Progressions - Common Ratio

Sum of first 12 terms of a GP is equal to the sum of the first 14 terms in the same GP. Sum of the first 17 terms is 92, what is the third term in the GP?
1. 92
2. -92
3. 46
4. 231
Choice A
92
CAT Progressions - Sum up to 'n' Terms

Sum of first 25 terms in AP is 525, sum of the next 25 terms is 725, what is the common difference?
1. \\frac{8}{25}\\)
2. \\frac{4}{25}\\)
3. \\frac{6}{25}\\)
4. \\frac{1}{25}\\)
Choice A
\\frac{8}{25}\\)
CAT Progressions - Common Difference

Let the n^th term of AP be defined as t_n, and sum up to 'n' terms be defined as S_n. If |t₈| = |t₁₆| and t₃ is not equal to t₇, what is S₂₃?
1. 23(t₁₆ - t₈)
2. 0
3. 23t₁₁
4. Cannot be determined
Choice B
0
CAT Progressions - Mean

a, b, c, d and e are 5 distinct numbers that from an arithmetic progression. They are not necessarily consecutive terms but form the first 5 terms of the AP. It is known that c is the arithmetic mean of a and b, and d is the arithmetic mean of b anc c. Which of the following statements are true?
i. Average of all 5 terms put together is c.
ii. Average of d and e is not greater than average of a and b.
iii. Average of b and c is greater than average of a and d.
1. i and ii only
2. ii and iii only
3. all three statements are true
4. i and iii only
Choice A
i and ii only
CAT Progressions - Median

Consider a, b, c in a G.P. such that |a + b + c| = 15. The median of these three terms is a, and b = 10. If a > c, what is the product of the first 4 terms of this G.P.?
1. 40000
2. 32000
3. 8000
4. 48000
Choice A
40000
CAT Progressions - Arithmetic Progressions

If 4 times the 4th term of an A.P. is equal to 9 times the 9th term of the A.P., what is 13 times the 13th term of this A.P.?
1. 7 times the 13th term
2. 0
3. 13 times the 7th term
4. 4 times the 4th term + 9 times the 9th term
Choice B
0
CAT Progressions - Arithmetic Progressions

Sequence P is defined by p_n = p_n-1 + 3, p₁ = 11, Sequence Q is defined as q_n = q_n-1 – 4, q₃ = 103. If p_k > q_k+2, what is the smallest value k can take?
1. 6
2. 11
3. 14
4. 15
Choice D
15
CAT Progressions - Arithmetic Progressions

The sum of 2n terms of A.P. {1, 5, 9, 13…..} is greater than sum of n terms of A.P. = {56, 58, 60..…}. What is the smallest value n can take?
1. 9
2. 10
3. 12
4. 14
Choice A
9
CAT Progressions - Arithmetic Progressions

a, b, c and d are in A.P., What can we say about terms bcd, acd, abd and abc?
1. They are also in A.P.
2. They are also in H.P.
3. They are also in G.P.
4. They are not in an A.P., G.P. or H.P.
Choice B
They are also in H.P.
CAT Progressions - Arithmetic Progressions

Second term in an AP is 8 and the 8th term is 2 more than thrice the second term. Find the sum up to 8 terms of this AP.
1. 124
2. 108
3. 96
4. 110
Choice A
124
CAT Progressions - Sum of infinite terms

If S_n = n³ + n² + n + 1 , where S_n denotes the sum of the first n terms of a series and t_m= 291, then m is equal to?
1. 24
2. 30
3. 26
4. 10
Choice D
10
CAT Progressions - Sum of infinite terms

Sum of infinite terms of a GP is 12. If the first term is 8, what is the 4th term of this GP?
1. \\frac{8}{27}\\)
2. \\frac{4}{27}\\)
3. \\frac{8}{20}\\)
4. \\frac{1}{3}\\)
Choice A
\\frac{8}{27}\\)
CAT Progressions - Sum of a Sequence

Find sum : 2² + 2 * 3² + 3 * 4² + 4 * 5².....10 * 11²
1. 6530
2. 3600
3. 2850
4. 3850
Choice D
3850
CAT Progressions - Ratio of Amounts

The salaries earned by two friends Anil and Jeetu in different years are in A.P. If the ratio of the amount earned by them in ‘p’ number of years are (4p+1) : (2p+17). Then find the ratio of amount earned by them in the 7^th year.
1. (2p+1) : (4p+6)
2. 53 : 43
3. 4 : 7
4. 15p : 36p
Choice B
53 : 43
CAT Progressions - Arithmetic Progressions

Ram invests a total sum of 2000 rupees on government bonds in 4 years. If these investments are in A.P and the sum of squares of the investments is 1200000. Find the investment made by ram in each year respectively. It is also known that he always invest more than the previous year.
1. 200,400,600,800
2. 875,625,375,125
3. 125,375,625,875
4. 50,350,650,950
Choice A
200,400,600,800
CAT Progressions - Arithmetic Progressions

Ram invest different amounts during the year on shares. S₁, S₂, S₃……….S_m are different sums of ‘n’ amounts invested in ‘m’ years. If the amounts invested during the years are in A.P whose first terms are 1,2,3…..m and common difference are 1,3,5…..,(2m-1) respectively then find the total amount invested by Ram in ‘m’ years.
1. n(m+1)
2. m+1
3. \\frac{mn}{2}\\)(mn+1)
4. cannot be determined
Choice C
\\frac{mn}{2}\\)(mn+1)
CAT Progressions - Sequence and Series

Find the sum of the series .4 + .44 + .444……. to n terms
1. 5.69
2. 14.44
3. \\frac{4}{81}\\)[9n-1+\\frac{1}{10^n}\\)]
4. \\frac{4}{81}\\)[n + 1]
Choice C
\\frac{4}{81}\\)[9n-1+\\frac{1}{10^n}\\)]
CAT Progressions - A.P, G.P and H.P

If the equation px² + 2qx + r = 0 and dx² + 2ex + f = 0 have a common root, and p,q,r are in G.P.,
then in which type of progression is \\frac{d}{p}\\) , \\frac{e}{q}\\) , \\frac{f}{r}\\)
1. \\frac{d}{p}\\) , \\frac{e}{q}\\) , \\frac{f}{r}\\) are in G.P
2. \\frac{d}{p}\\) , \\frac{e}{q}\\) , \\frac{f}{r}\\) are in A.P
3. \\frac{d}{p}\\) , \\frac{e}{q}\\) , \\frac{f}{r}\\) are in H.P
4. Insufficient Data
Choice B
\\frac{d}{p}\\) , \\frac{e}{q}\\) , \\frac{f}{r}\\) are in A.P
CAT Progressions - Sum of all Terms

Find the sum of all the terms, If the first 3 terms among 4 positive 2-digit integers are in A.P and the last 3 terms are in G.P. Moreover the difference between the first and last term is 40.
1. 108
2. 172
3. 124
4. 196
Choice B
172

The Questions that follow, are from actual CAT papers. If you wish to take them separately or plan to solve actual CAT papers at a later point in time, It would be a good idea to stop here.

CAT 2023 Slot 3 - QA

The value of \(1+\left(1+\frac{1}{3}\right) \frac{1}{4}+\left(1+\frac{1}{3}+\frac{1}{9}\right) \frac{1}{16}+\left(1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}\right) \frac{1}{64}+\cdots\), is
1. \(\frac{15}{13}\)
2. \(\frac{16}{11}\)
3. \(\frac{27}{12}\)
4. \(\frac{15}{8}\)
Choice B
\(\frac{16}{11}\)
CAT 2023 Slot 3 - QA

Let \(a_n=46+8 n\) and \(b_n=98+4 n\) be two sequences for natural numbers \(n \leq 100\). Then, the sum of all terms common to both the sequences is
1. 15000
2. 14900
3. 14602
4. 14798
Choice B
14900
CAT 2023 Slot 2 - QA

Let both the series \(a_1, a_2, a_3, \ldots\) and \(b_1, b_2, b_3 \ldots\) be in arithmetic progression such that the common differences of both the series are prime numbers. If \(a_5=b_9, a_{19}=b_{19}\) and \(b_2=0\), then \(a_{11}\) equals
1. 79
2. 83
3. 86
4. 84
Choice A
79
CAT 2023 Slot 1 - QA

For some positive and distinct real numbers \(x, y\) and \(z\), if \(\frac{1}{\sqrt{y}+\sqrt{z}}\) is the arithmetic mean of \(\frac{1}{\sqrt{x}+\sqrt{z}}\) and \(\frac{1}{\sqrt{x}+\sqrt{y}}\), then the relationship which will always hold true, is
1. \(\sqrt{x}, \sqrt{y}\) and \(\sqrt{z}\) are in arithmetic progression
2. \(\sqrt{x}, \sqrt{z}\) and \(\sqrt{y}\) are in arithmetic progression
3. \(y, x\) and \(z\) are in arithmetic progression
4. \(x, y\) and \(z\) are in arithmetic progression
Choice C
\(y, x\) and \(z\) are in arithmetic progression
CAT 2023 Slot 1 - QA

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 \(n^{\text {th }}\) day exceeds one million, then the lowest possible value of \(n\) is
19
CAT 2022 Slot 2 - QA

On day one, there are 100 particles in a laboratory experiment. On day \(n\), where \(n \geq 2\), 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
1. 19
2. 16
3. 17
4. 18
Choice A
19
CAT 2022 Slot 2 - QA

Consider the arithmetic progression \(3,7,11, \ldots\) and let \(A_n\) denote the sum of the first \(n\) terms of this progression. Then the value of \(\frac{1}{25} \sum_{n=1}^{25} A_n\) is
1. 404
2. 442
3. 455
4. 415
Choice C
455
CAT 2022 Slot 1 - QA

For any natural number \(n\), suppose the sum of the first \(n\) terms of an arithmetic progression is \(\left(n+2 n^2\right)\). If the \(n^{\text {th }}\) term of the progression is divisible by 9 , then the smallest possible value of \(n\) is
1. 4
2. 8
3. 7
4. 9
Choice C
7
CAT 2021 Slot 3 - QA

Consider a sequence of real numbers \(x_{1}, x_{2}, x_{3}, \ldots\) such that \(x_{n+1}=x_{n}+n-1\) for all \(n \geq 1 .\) If \(x_{1}=-1\) then \(x_{100}\) is equal to
1. 4949
2. 4849
3. 4850
4. 4950
Choice C
4850

Correct: 10.44%
Incorrect: 18.71%
Unattempted: 70.85%
CAT 2021 Slot 2 - QA

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
1. 8
2. 10
3. 14
4. 12
Choice C
14

Correct: 22.54%
Incorrect: 10.92%
Unattempted: 66.54%
CAT 2021 Slot 2 - QA

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
1. 2
2. -2
3. 200
4. -200
Choice B
-2

Correct: 13.8%
Incorrect: 14.98%
Unattempted: 71.22%
CAT 2021 Slot 1 - QA

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?
1. 4
2. 3
3. 1
4. 2
Choice D
2

Correct: 19.61%
Incorrect: 26.55%
Unattempted: 53.84%
CAT 2020 Question Paper Slot 2 - Sequence & series

Let teh m-th and n-thterms of a Geometric progression be \\frac{3}{4}) and 12, respectively, when m < n. If the common ratio of the progression is an integer r, then the smallest possible value of r + n - m is
1. -4
2. -2
3. 6
4. 2
CAT 2020 Question Paper Slot 1 - Sequence ∧ Series

A gentleman decided to treat a few children in the following manner. He gives half of his total stock of toffees and one extra to the first child, and then the half of the remaining stock along with one extra to the second and continues giving away in this fashion. His total stock exhausts after he takes care of 5 children. How many toffees were there in his stock initially?

Correct Answer: 62
CAT 2019 Question Paper Slot 2 - Sequence & series

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
1. -1
2. 1
3. 0
4. 10
Choice B
1
CAT 2019 Question Paper Slot 2 - - Sequence & series

The number of common terms in the two sequences: 15, 19, 23, 27, ...... , 415 and 14, 19, 24, 29, ...... , 464 is
1. 20
2. 18
3. 21
4. 19
Choice A
20
CAT 2019 Question Paper Slot 2 - Sequence & series

If (2n+1) + (2n+3) + (2n+5) + ... + (2n+47) = 5280 , then what is the value of 1+2+3+ ... +n ? [TITA]
4851
CAT 2019 Question Paper Slot 1 - Progressions

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
1. (1003)¹⁵ + 6
2. (977)¹⁵ - 3
3. (1003)2¹⁵ - 3
4. (977)2¹⁴ + 3
Choice C
(1003)2¹⁵ - 3
CAT 2019 Question Paper Slot 1 - Progressions

If a₁ + a₂ + a₃ + ... + a_n = 3(2ⁿ⁺¹ - 2), then a₁₁ equals [TITA]
6144
CAT 2019 Question Paper Slot 1 - Progressions

If a₁, a₂, ......... are in A.P , \\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}}}) then , is equal to
1. \\frac{n}{\sqrt{a_1} + \sqrt{a_{n+1}}})
2. \\frac{n - 1}{\sqrt{a_1} + \sqrt{a_n}})
3. \\frac{n}{\sqrt{a_1} - \sqrt{a_{n+1}}})
4. \\frac{n - 1}{\sqrt{a_1} + \sqrt{a_{n-1}}})
Choice A
\\frac{n}{\sqrt{a_1} + \sqrt{a_{n+1}}})
CAT 2018 Question Paper Slot 2 - Sequence & Series

Let a₁, a₂, ... , a₅₂ be positive integers such that a₁ ＜ a₂ ＜ ... ＜ a₅₂. Suppose, their arithmetic mean is one less than the arithmetic mean of a₂, a₃, ..., a₅₂. If a₅₂ = 100, then the largest possible value of a₁ is
1. 48
2. 20
3. 45
4. 23
Choice D
23
CAT 2018 Question Paper Slot 2 - Sequence & Series

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 (TITA)
24
CAT 2018 Question Paper Slot 2 - Sequence & Series

The value of the sum 7 x 11 + 11 x 15 + 15 x 19 + ..... + 95 x 99 is
1. 80707
2. 80751
3. 80730
4. 80773
Choice A
80707
CAT 2018 Question Paper Slot 2 - Progressions & Series

The arithmetic mean of x, y and z is 80, and that of x, y, z, u and v is 75, where u = \\frac{(x+y)}{2}\\) and v = \\frac{(y+z)}{2}\\). If x ≥ z, then the minimum possible value of x is (TITA)
105
CAT 2018 Question Paper Slot 1 - Sequence & Series

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
1. \\frac{1}{6})
2. \\frac{3}{6})
3. \\frac{3}{2})
4. \\frac{5}{2})
Choice D
\\frac{5}{2})
CAT 2017 Question Paper Slot 2 - Progressions

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 [TITA]
51
CAT 2017 Question Paper Slot 2 - Progressions

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
1. \\frac{1}{32})
2. \\frac{2}{32})
3. \\frac{3}{32})
4. \\frac{4}{32})
Choice C
\\frac{3}{32})
CAT 2017 Question Paper Slot 2 - Progressions

If a₁ = \\frac{1}{2 × 5}) , a₂ = \\frac{1}{5 × 8}) , a₃ = \\frac{1}{8 × 11}),...., then a₁ + a₂ + a₃ + ...... + a₁₀₀ is
1. \\frac{25}{151})
2. \\frac{1}{2})
3. \\frac{1}{4})
4. \\frac{111}{55})
Choice A
\\frac{25}{151})
CAT 2017 Question Paper Slot 1 - Progressions

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:
1. 2 : 3
2. 3 : 2
3. 3 : 4
4. 4 : 3
Choice A
2 : 3
CAT 2017 Question Paper Slot 1 - Progressions

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?
1. 8
2. 9
3. 10
4. 11
Choice B
9

The Questions that follow, are from actual XAT papers. If you wish to take them separately or plan to solve actual XAT papers at a later point in time, It would be a good idea to stop here.

XAT 2019 Question Paper - QADI
When opening his fruit shop for the day a shopkeeper found that his stock of apples could be perfectly arranged in a complete triangular array: that is, every row with one apple more than the row immediately above, going all the way up ending with a single apple at the top. During any sales transaction, apples are always picked from the uppermost row, and going below only when that row is exhausted.
When one customer walked in the middle of the day she found an incomplete array in display having 126 apples totally. How many rows of apples (complete and incomplete) were seen by this customer? (Assume that the initial stock did not exceed 150 apples.)
1. 15
2. 14
3. 13
4. 12
5. 11
Choice D
12
XAT 2018 Question Paper - QADI
An antique store has a collection of eight clocks. At a particular moment, the displayed times on seven of the eight clocks were as follows: 1:55 pm, 2:03 pm, 2:11 pm, 2:24 pm, 2:45 pm, 3:19 pm and 4:14 pm. If the displayed times of all eight clocks form a mathematical series, then what was the displayed time on the remaining clock?
1. 1:53 pm
2. 1:58 pm
3. 2:18 pm
4. 3:08 pm
5. 5:08 pm
Choice B
1:58 pm
XAT 2018 Question Paper - QADI
David has an interesting habit of spending money. He spends exactly £X on the Xth day of a month. For example, he spends exactly £5 on the 5th of any month. On a few days in a year, David noticed that his cumulative spending during the last 'four consecutive days' can be expressed as 2N where N is a natural number. What can be the possible value(s) of N?
1. 5
2. 6
3. 7
4. 8
5. n can have more than one value
Choice B
6

The Questions that follow, are from actual IPMAT papers. If you wish to take them separately or plan to solve actual IPMAT papers at a later point in time, It would be a good idea to stop here.

IPMAT 2020 Sample Paper - IPM Rohtak Quants
Fruits were purchased for Rs 350. 9 boys ate \\frac{3}{5}\\)^th of them in 2 hours. 6 boys feel their stomach as full so do not eat further. In how many hours the remaining fruits will get finished by remaining boys?
1. 2 hours
2. 3 hours
3. 5 hours
4. 4 hours
Choice D
4 hours
IPMAT 2020 Sample Paper - IPM Rohtak Quants
What should come at the place of question mark? 46080, 3840, 384, 48, 8, 2, ?
1. 1
2. \\frac{1}{64}\\)
3. \\frac{1}{8}\\)
4. None of these
Choice A
1
IPMAT 2020 Sample Paper - IPM Rohtak Quants
The sum of third and ninth term of an A.P is 8. Find the sum of the first 11 terms of the progression.
1. 44
2. 22
3. 19
4. None of the above
Choice A
44
IPMAT 2020 Question Paper - IPM Rohtak Quants
The height of nineteen people of comic book is in Arithmetic progression. The average height of them is 19 feet. If the tallest is 37 feet. Then what is the weight of the shortest?
1. 2
2. 1
3. 3
4. 4
Choice B
1
IPMAT 2019 Question Paper - IPM Indore Quants
If (1 + x - 2x²)⁶ = \A_{0}+\sum_{r=1}^{12} A_{r} x^{r}\\), then value of \A_{2}+A_{4}+A_{6}+\cdots+A_{12}\\) is
1. 31
2. 32
3. 30
4. 29
Choice A
31
IPMAT 2019 Question Paper - IPM Indore Quants
The number of terms common to both the arithmetic progressions 2,5,8,11,...., 179 and 3,5,7,9,....., 101 is
1. 17
2. 16
3. 19
4. 15
Choice A
17
IPMAT 2019 Question Paper - IPM Indore Quants
There are numbners \a_{1}, a_{2}, a_{3}, \ldots, a_{n}\\) each of them being +1 or -1. If it is known that \a_{1} a_{2} + a_{2} a_{3} + a_{3} a_{4} + \ldots a_{n-1} a_{n} + a_{n} a_{1} = 0\\) then
1. n is a multiple of 2 but not a multiple of 4
2. n is a multiple of 3
3. n can be any multiple of 4
4. The only possible value of n is 4
Choice C
n can be any multiple of 4