# Top 31 CAT Number Series Questions With Video Solutions

You can find all the Top 30+ CAT Number Series questions from the previous papers with detailed video explanations on this page. The CAT Number Series plays a crucial role in CAT quantitative section. There are many tricks, shortcuts and formulas that help you to solve the questions quickly. One can find those solving tips in the video solutions explained by CAT experts and IIM Alumni. Click on the link below to download all the number system questions from CAT previous papers PDF.  Keep practising free CAT mocks where you'll get a fair idea of how questions are asked, and type of questions asked.

## CAT Number Series Formulas

1. A.G.P. Properties Formula

Arithmetic Geometric Series

A series will be an arithmetic-geometric series if each of its terms is formed by the product of the corresponding terms of an A.P and G.P.

The general form of A.G.P series is a, (a+d)r, (a+2d)$$r^{2}$$,......

Sum of ‘n’ terms of A.G.P series

$$S_{n}$$=$$\frac{a}{1-r}$$+rd$$\frac{(1-r^{n-1})}{1-r}$$+rn$$\frac{[a+(n-1)d]}{1-r}$$(r≠1)

Sum of infinite terms of A.G.P series

$$S_{∞}$$=$$\frac{a}{1-r}+\frac{dr}{(1-r)^{2}}$$(|r|<1)

2. G.P. - Formulas and Properties

Geometric Progression

If in a succession of numbers the ratio of any term and the previous term is constant then that numbers are said to be in Geometric Progression.

Ex :1, 3, 9, 27 or a, ar, a$$r^{2}$$, a$$r^{3}$$

The general expression of a G.P, Tn = a $$r^{n-1}$$ (where a is the first term and ‘r’ is the common ratio).

Sum of ‘n’ terms in G.P, Sn = $$\frac{a(1-r^{n})}{1-r}$$ (if r<1) or $$\frac {a(r^{n}-1)}{r-1}$$ (if r>1)

Properties of G.P

If a, b , c, d,.... are in G.P and ‘k’ is a constant then

1. ak, bk, ck,...will also be in G.P
2. a/k, b/k, c/k will also be in G.P

Sum of term of infinite series in G.P, $$S_{∞}$$=$$\frac {a}{1-r}$$ (-1 < r <1)

3. A.P. - Formulas and Properties

Arithmetic progression (A.P)

If the sum of the difference between any two consecutive terms is constant then the terms are said to be in A.P

Example: 2,5,8,11 or a, a+d, a+2d, a+3d...

If 'a' is the first term and 'd' is a common difference then the general 'n' term is $$T_{n}$$=a+(n-1)d

Sum of first 'n' terms in A.P=$$\frac{n}{2}$$[2a+(n-1)d]

Number of terms in A.P=$$\frac{Last Term-First Term}{Common Difference}$$+1

Properties of Arithmetic progression

If a, b, c, d,.... are in A.P and ‘k’ is a constant then

a-k, b-k, c-k,... will also be in A.P

ak, bk, ck,...will also be in A.P

a/k, b/k, c/k will also be in A.P

4. Harmonic Mean Formula

Harmonic Mean

If a, b, c, d...are the given numbers in H.P then the Harmonic mean of 'n' terms=$$\frac{Number of terms}{\frac{1}{a}+\frac{1}{a}+\frac{1}{a}+....}$$

If two numbers a and b are in H.P then the Harmonic mean= $$\frac{2ab}{a+b}$$

## CAT 2023 Number Series questions

#### Question 1

Let both the series $$a_{1},a_{2},a_{3}$$... and $$b_{1},b_{2},b_{3}$$... 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

#### Question 2

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

#### Question 3

Let $$a_n = 46 + 8n$$ and $$b_n = 98 + 4n$$ be two sequences for natural numbers $$n \leq 100$$. Then, the sum of all terms common to both the sequences is

#### Question 4

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 5

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 2022 Number Series questions

#### Question 1

The average of a non-decreasing sequence of N numbers $$a_{1},a_{2}, ... , a_{N}$$ is 300. If $$a_1$$, is replaced by $$6a_{1}$$ , the new average becomes 400. Then, the number of possible values of $$a_{1 }$$, is

#### Question 2

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

#### Question 3

On day one, there are 100 particles in a laboratory experiment. On day n, where $$n\ge2$$, one out of every n articles produces another particle. If the total number of particles in the laboratory experiment increases to 1000 on day m, then m equals

#### Question 4

Consider the arithmetic progression 3, 7, 11, ... 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

## CAT 2021 Number Series questions

#### Question 1

Consider a sequence of real numbers, $$x_{1},x_{2},x_{3},...$$ such that $$x_{n+1}=x_{n}+n-1$$ for all $$n\geq1$$. If $$x_{1}=-1$$ then $$x_{100}$$ is equal to

#### Question 2

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

#### Question 3

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

#### Question 4

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

## CAT 2020 Number Series questions

#### Question 1

If $$x_1=-1$$ and $$x_m=x_{m+1}+(m+1)$$ for every positive integer m, then $$X_{100}$$ equals

## CAT 2019 Number Series questions

#### Question 1

Let $$a_1, a_2, ...$$ be integers such that
$$a_1 - a_2 + a_3 - a_4 + .... + (-1)^{n - 1} a_n = n,$$ for all $$n \geq 1.$$
Then $$a_{51} + a_{52} + .... + a_{1023}$$ equals

#### Question 2

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

#### Question 3

If $$a_1 + a_2 + a_3 + .... + a_n = 3(2^{n + 1} - 2)$$, for every $$n \geq 1$$, then $$a_{11}$$ equals

#### Question 4

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

## CAT 2018 Number Series questions

#### Question 1

The value of the sum 7 x 11 + 11 x 15 + 15 x 19 + ...+ 95 x 99 is

## CAT 2017 Number Series questions

#### Question 1

If $$a_{1}=\frac{1}{2\times5},a_{2}=\frac{1}{5\times8},a_{3}=\frac{1}{8\times11},...,$$ then $$a_{1}+a_{2}+a_{3}+...+a_{100}$$ is