Progressions

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

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?

A

x, y and z are in Arithmetic Progression

B

y, x and z are in Arithmetic Progression

C

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

D

√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?

A

86

B

84

C

79

D

83

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

A

14798

B

14602

C

14900

D

15000

Question 6.

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

A

15/13

B

27/12

C

16/11

D

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

A

9

B

8

C

4

D

7

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

A

415

B

404

C

455

D

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.

A

16

B

17

C

19

D

18

Question 10.

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

Instructions

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 11.

A

1

B

3

C

4

D

2

Instructions

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

Question 12.

A

4941

B

6119

C

7471

D

6090

Instructions

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.

Question 13.

A

-2

B

2

C

-200

D

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

A

12

B

8

C

14

D

10

Instructions

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

Question 15.

A

4850

B

4849

C

4950

D

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

A

6

B

2

C

-2

D

-2

Instructions

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

Question 17.

A

-5151

B

-5150

C

-5050

D

-5051

Instructions

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

Question 18.

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

A

(1003)^15 + 6

B

(997)^15 - 3

C

(1003)*2^(15) - 3

D

(997)*2^(14) + 3

Instructions

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

Question 20.

A

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

B

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

C

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

D

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

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Question 1.

Question 2.

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

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 ana_{n}an​ and bnb_{n}bn​ be two sequences such that ana_{n}an​ = 13 + 6(n - 1) and bnb_{n}bn​ = 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 ana_{n}an​ = 46 + 8n and bnb_{n}bn​ = 98 + 4n be two sequences for natural numbers n ≤ 100. Then, the sum of all terms common to both the sequences is

Question 6.

The value of 1 +(1+13)14\left( 1+\frac{1}{3} \right)\frac{1}{4}(1+31​)41​+(1+13+19)116\left( 1+\frac{1}{3}+\frac{1}{9} \right)\frac{1}{16}(1+31​+91​)161​+(1+13+19+127)164\left( 1+\frac{1}{3}+\frac{1}{9} +\frac{1}{27}\right)\frac{1}{64}(1+31​+91​+271​)641​

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 125∑n=125An\frac{1}{25}\sum_{n=1}^{25}A_{n}251​n=1∑25​An​

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

Instructions

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

Question 11.

Instructions

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 15th group is equal to

Question 12.

Instructions

For a sequence of real numbers x1, x2, …, xn, if x1 - x2 + x3 - … + (-1)(n+1) xn = n2 + 2n for all natural numbers n, then the sum x49 + x50 equals.

Question 13.

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

Instructions

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

Question 15.

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

Instructions

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

Question 17.

Instructions

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

Question 18.

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

Instructions

If a1, a2, ... are in A.P., then, 1a1+a2\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}a1​​+a2​​1​ + 1a2+a3\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}a2​​+a3​​1​ + ... + 1an+an+1\frac{1}{\sqrt{a_{n}}+\sqrt{a_{n+1}}}an​​+an+1​​1​ is equal to

Question 20.

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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?

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

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

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

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.

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

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 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