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.
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. P_{6} > P_{5} and P_{6} > P_{7}, what is the sum of all possible values of n?
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?
Sum of first 25 terms in AP is 525, sum of the next 25 terms is 725, what is the 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 |t_{8}| = |t_{16}| and t_{3} is not equal to t_{7}, what is S_{23}?
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.
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.?
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.?
Sequence P is defined by p_{n} = p_{n-1} + 3, p_{1} = 11, Sequence Q is defined as q_{n} = q_{n-1} – 4, q_{3} = 103. If p_{k} > q_{k+2}, what is the smallest value k can take?
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?
a, b, c and d are in A.P., What can we say about terms bcd, acd, abd and abc?
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.
If S_{n} = n^{3} + n^{2} + 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?
Sum of infinite terms of a GP is 12. If the first term is 8, what is the 4th term of this GP?
Find sum : 2^{2} + 2 * 3^{2} + 3 * 4^{2} + 4 * 5^{2}.....10 * 11^{2}
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.
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.
Ram invest different amounts during the year on shares. S_{1}, S_{2}, S_{3}……….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.
Find the sum of the series .4 + .44 + .444……. to n terms
If the equation px^{2} + 2qx + r = 0 and dx^{2} + 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}\\)
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.
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.
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
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
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
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
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
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
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
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
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
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
For a sequence of real numbers x_{1}, x_{2}, ..., x_{n}, if x_{1} - x_{2} + x_{3} - ... + (-1)^{n + 1}x_{n} = n^{2} + 2n for all natural numbers n, then the sum x_{49} + x_{50} equals
If x_{0} = 1, x_{1} = 2, and x_{n + 2} = \(\frac{1+x_{n+1}}{x_{n}}\), n = 0, 1, 2, 3,..., then x_{2021} is equal to?
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
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?
Let a_{1} , a_{2} be integers such that a_{1} - a_{2} + a_{3} - a_{4} + ........ +(-1)^{n-1} a_{n} = n , for n ≥ 1. Then a_{51} + a_{52} + ........ + a_{1023} 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 ? [TITA]
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 a_{1} + a_{2} + a_{3} + ... + a_{n} = 3(2^{n+1} - 2), then a_{11} equals [TITA]
If a_{1}, a_{2}, ......... 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
Let a_{1}, a_{2}, ... , a_{52} be positive integers such that a_{1} ＜ a_{2} ＜ ... ＜ a_{52}. Suppose, their arithmetic mean is one less than the arithmetic mean of a_{2}, a_{3}, ..., a_{52}. If a_{52} = 100, then the largest possible value of a_{1} is
Let t_{1}, t_{2},… be real numbers such that t_{1}+ t_{2} +... + t_{n} = 2n^{2} + 9n + 13, for every positive integer n ≥ 2. If t_{k}=103, then k equals (TITA)
The value of the sum 7 x 11 + 11 x 15 + 15 x 19 + ..... + 95 x 99 is
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)
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
Let a_{1}, a_{2}, a_{3}, a_{4}, a_{5} be a sequence of five consecutive odd numbers. Consider a new sequence of five consecutive even numbers ending with 2a_{3}. If the sum of the numbers in the new sequence is 450, then a_{5} is [TITA]
An infinite geometric progression a_{1}, a_{2}, a_{3},... has the property that a_{n} = 3(a_{n+1} + a_{n+2} +....) for every n ≥ 1. If the sum a_{1} + a_{2} + a_{3} +...... = 32, then a_{5} is
If a_{1} = \\frac{1}{2 × 5}) , a_{2} = \\frac{1}{5 × 8}) , a_{3} = \\frac{1}{8 × 11}),...., then a_{1} + a_{2} + a_{3} + ...... + a_{100} is
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 a_{1}, a_{2},.......a_{3n} be an arithmetic progression with a_{1} = 3 and a_{2} = 7. If a_{1} + a_{2} + ......+a_{3n} = 1830, then what is the smallest positive integer m such that m (a_{1} + a_{2} + ..... + a_{n}) > 1830?
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.
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.)
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?
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?
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.
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?
What should come at the place of question mark? 46080, 3840, 384, 48, 8, 2, ?
The sum of third and ninth term of an A.P is 8. Find the sum of the first 11 terms of the progression.
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?
If (1 + x - 2x^{2})^{6} = \A_{0}+\sum_{r=1}^{12} A_{r} x^{r}\\), then value of \A_{2}+A_{4}+A_{6}+\cdots+A_{12}\\) is
The number of terms common to both the arithmetic progressions 2,5,8,11,...., 179 and 3,5,7,9,....., 101 is
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
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