Progressions

Test Name	No. of Questions	Marks (of each)	Time	Take Test
Test-1	10	3	40 Minutes	Take Test
Test-2	10	3	40 Minutes	Take Test

Tips To Solve CAT Progressions

- Fundamentals of this concept are useful in solving the questions of the other topics by assuming the unknown values as variables. Make sure to cover other inter-related concepts of CAT syllabus. All the inter-related concepts need to be covered to have a good foundation in concepts.

- Be careful of silly mistakes in this topic, as that is how students generally lose marks here. The number of equations needed to solve the given problem equals the number of variables. A linear equation is an equation which gives a straight line when plotted on a graph.

- If you are confused, enrolling in CAT online coaching will help you a long way.

- Linear equations can be of one variable or two variables, or three variables.

Let a, b, c and d be constants, and x, y, and z are variables. A general form of a single variable linear equation is ax + b = 0.
A general form of two-variable linear equation is ax + by = c.
A general form of three-variable linear equation is ax + by + cz = d.

CAT Progressions PDF

To help CAT aspirants in their preparation, we have made a comprehensive formula PDF containing all the important linear equations that are essential. This PDF includes all the necessary formulas, techniques, and examples required to solve linear equations efficiently. Click on the link below to download the Linear equations formula PDF.

1. Linear Equations Formulae: Solving Linear Equations

For equations of the form ax + by = c and mx + ny = p, find the LCM of b and n.

Multiply each equation with a constant to make the y term coefficient equal to the LCM. Then subtract equation 2 from equation 1.

2. Linear Equations Formulae: Straight Lines

Equations with 2 variables: Consider two equations ax + by = c and mx + ny = p. Each of these equations represents two lines on the x-y coordinate plane. The solution of these equations is the point of intersection.

If $\frac{a}{m} = \frac{b}{n} \neq \frac{c}{p}$ : This means that both the equations have the same slope but different intercepts, and hence are parallel to each other. There is no point of intersection and no solution.

If $\frac{a}{m} \neq \frac{b}{n}$ : They have different slopes and hence must intersect at some point, resulting in a unique solution.

If $\frac{a}{m} = \frac{b}{n} = \frac{c}{p}$ : The two lines have the same slope and intercept. Hence, they are the same lines. As they have infinite points common between them, there are infinitely many solutions possible.

Question 1.

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 nth day exceeds one million, then the lowest possible value of n is

Question 2.

For some positive and distinct real numbers x, y and z if $\frac{1}{\sqrt{y}+\sqrt{x}}$ 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?

x, y and z are in Arithmetic Progression

y, x and z are in Arithmetic Progression

√x, √y and √z are in Arithmetic Progression

√y, √x and √z are in Arithmetic Progression

Question 3.

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?

Question 4.

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

Question 5.

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

14798

14602

14900

15000

Question 6.

The value of 1 + $\left( 1+\frac{1}{3} \right)\frac{1}{4}$ + $\left( 1+\frac{1}{3}+\frac{1}{9} \right)\frac{1}{16}$ + $\left( 1+\frac{1}{3}+\frac{1}{9} +\frac{1}{27}\right)\frac{1}{64}$

15/13

27/12

16/11

15/8

Question 7.

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

Question 8.

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 $\frac{1}{25}\sum_{n=1}^{25}A_{n}$

415

404

455

442

Question 9.

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.

Question 10.

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

Question 11.

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

Question 12.

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

4941

6119

7471

6090

Question 13.

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.

-2

-200

200

Question 14.

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

Question 15.

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

4850

4849

4950

4949

Question 16.

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

-2

Question 17.

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

-5151

-5150

-5050

-5051

Question 18.

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

Question 19.

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

(1003)^15 + 6

(997)^15 - 3

(1003)*2^(15) - 3

(997)*2^(14) + 3

Question 20.

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

\frac{n}{\sqrt{a_{1}}-\sqrt{a_{n+1}}}

\frac{n}{\sqrt{a_{1}}+\sqrt{a_{n+1}}}

\frac{n-1}{\sqrt{a_{1}}+\sqrt{a_{n-1}}}

\frac{n-1}{\sqrt{a_{1}}+\sqrt{a_{n}}}

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