Mathematics · Quantitative Aptitude

Sequences and Series

230 Questions

Sequences and series involve ordered lists of numbers and the sum of their terms. The questions primarily test knowledge of arithmetic progressions, geometric progressions, and infinite series. It is an essential part of quantitative aptitude that requires strong pattern recognition skills.

Arithmetic progressionGeometric progressionInfinite seriesSum of termsNumber sequences

Sequences and Series Questions

Multiple choice median percentiles and quartiles range and mean deviation mode

If the 2nd term of a GM series is 25, the first and 3rd terms will be _____.

  1. 5,125

  2. 105,5

  3. 5,105

  4. 25,125
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Geometric mean between two numbers is the mean proportion of the series. 

Therefore, let the two numbers be a and b 
a/25 = 25/b 

=>ab= 25 x 25

Also , a/b = 1/25

 b = 25a

so, a (25a) = 25 x 25

=>a2 = 25

=> a = 5

Now, b = 25(5)

        b = 125  

Therefore, the first and the 3rd terms are 5 and 125 respectively. 

Multiple choice telescopic summation for infinte series binomial theorem, sequence and series maths

Find the sum of the following geometric series:
$ \sqrt{7}, \sqrt{21}, 3\sqrt{7},...$ to n terms

  1. $ \sqrt{7}\left ( \dfrac{3^{-n/2}-1}{\sqrt{3}-1} \right )$
  2. $ \sqrt{6}\left ( \dfrac{3^{n/2}-1}{\sqrt{3}-1} \right )$
  3. $ \sqrt{7}\left ( \dfrac{3^{n/2}-1}{\sqrt{3}-1} \right )$
  4. $ \sqrt{5}\left ( \dfrac{3^{n/2}-1}{\sqrt{3}-1} \right )$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let $ S _n$ denote the sum of n terms of the G.P. $\sqrt{7}, \sqrt{21}, 3\sqrt{7}, ...,$ 


Cleraly, the given series is a $G.P.$ with first term$=a=\sqrt7$ and common ratio$=r=\sqrt 3$

Then,
$ S _n = \sqrt{7} \left { \dfrac{\left ( \sqrt{3} \right )^{n}-1}{\sqrt{3}-1} \right } = \sqrt{7} \left ( \dfrac{3^{n/2}-1}{3^{1/2}-1} \right )$

Multiple choice telescopic summation for infinte series binomial theorem, sequence and series maths

Find the sum of the first 100 terms -5, -4, -3, -2, -1, 0, 1, 2 ............. using Gauss method

  1. 4,400

  2. 4,100

  3. 4,200

  4. 4,450
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given that $a = -5, n = 100, a _n =?, d = 1$
we know that Gauss formula is, $S _n = \dfrac{n}{2}$   [First term + Last term]
To find nth term,
$a _n = a + (n - 1) d$
$a _{100} = - 5 + (100 - 1)1$
$= - 5 + 99$
$a _{100} =94$
$S _n = \dfrac{100}{2} [-5 + 94]$
$= 50 [89]$
$S _{100} = 4,450$

Multiple choice telescopic summation for infinte series binomial theorem, sequence and series maths

Find the sum of first 31 terms of an A.P. whose third term is 12 and fourth term is 16.

  1. 1,983

  2. 1,984

  3. 1,985

  4. 1,986
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given that, $a _3 = 12; a _4 = 16$
Common difference, $d = a _4 - a _3 = 16 - 12 = 4$
$a _3 - a _2 = d$
$12 - 4 = a _2 $ $\Rightarrow  8$
$d = a _2 - a _1$ 
$a = 4$
We know the formula,
$s _n = \dfrac{n}{2} [2a + (n - 1)d]$
$S _{31} = \dfrac{31}{2} [2 \times 4 + (31 - 1)4]$
$= 15.5 [8 + 30 \times 4]$
$=15.5 [128]$
$S _{31} = 1,984$

Multiple choice telescopic summation for infinte series binomial theorem, sequence and series maths

The sum of the first 12 terms is 100. The first term is 20. Find the last term. (use Gauss method)

  1. $\dfrac{-20}{6}$
  2. $\dfrac{-10}{6}$
  3. $\dfrac{-15}{5}$
  4. $\dfrac{-30}{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given that $S _n = 100, n = 12, a = 20$. fast term = ?
we know that, $S _n = \dfrac{n}{2}$  [First term + Last term]
$100 = \dfrac{12}{2}$   [20 + Last term]
$100 = 120 + 6 (\text{Last term})$
Last term $= \dfrac{-20}{6}$

Multiple choice telescopic summation for infinte series binomial theorem, sequence and series maths

The sum of infinity of the series $\displaystyle 1+\frac{4}{5}+\frac{7}{5^{2}}+\frac{10}{5^{3}}+$..... is

  1. $\displaystyle\frac{16}{35}$
  2. $\displaystyle\frac{11}{8}$
  3. $\displaystyle\frac{35}{16}$
  4. $\displaystyle\frac{8}{6}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let $\displaystyle S=1+\frac { 4 }{ 5 } +\frac { 7 }{ { 5 }^{ 2 } } +\frac { 10 }{ { 5 }^{ 3 } } +...$   ...(1)
Multiply (1) by $\displaystyle \frac { 1 }{ 5 } $ , we get

$\displaystyle \frac { 1 }{ 5 } S=\frac { 1 }{ 5 } +\frac { 4 }{ { 5 }^{ 2 } } +\frac { 7 }{ { 5 }^{ 3 } } +\frac { 10 }{ { 5 }^{ 4 } } +...$   ...(2)

$(1) -(2)$, gives 
$\displaystyle \left( 1-\frac { 1 }{ 5 }  \right) S=1+\frac { 3 }{ 5 } +\frac { 3 }{ { 5 }^{ 2 } } +\frac { 3 }{ { 5 }^{ 3 } } +...$

$\displaystyle \Rightarrow \frac { 4 }{ 5 } S=1+\frac { 3 }{ 5 } \left( 1+\frac { 1 }{ 5 } +\frac { 1 }{ { 5 }^{ 2 } } +... \right) $

$\displaystyle \Rightarrow \dfrac { 4 }{ 5 } S=1+\dfrac { 3 }{ 5 } \left( \dfrac { 1 }{ 1-\dfrac { 1 }{ 5 }  }  \right) \Rightarrow \dfrac { 4 }{ 5 } S=1+\dfrac { 3 }{ 5 } \left( \dfrac { 5 }{ 4 }  \right) $

$\displaystyle \Rightarrow \frac { 4 }{ 5 } S=1+\frac { 3 }{ 4 } \Rightarrow S=\frac { 35 }{ 16 } $

Multiple choice telescopic summation for infinte series binomial theorem, sequence and series maths

Find the sum of the first 25 terms of the A.P.: 2 + 5 + 8 + 11 + ............ (use Gauss method)

  1. 910

  2. 930

  3. 950

  4. 940
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given that $a = 2; d = 3$
$n = 25 ; a _{25} = $?
Using Gauss method, we find the value of $a _{25}$
$a _n = a + (n - 1) d$
$a _{25} = 2 + (25 - 1) 3$
$= 2 + (24) 3$
$a _{25} = 74$
Sum of 'n' series using Gauss method is,
$S _n = \dfrac{n}{2} $   [First term + Last term]
$= \dfrac{25}{2} [2 + 74]$
$= 12.5 (76)$
$S _n = 950$

Multiple choice maths numbers and sequences series introduction to series introduction to sequences and series

When each term of a sequence is connected using $a +$ or $a -$ sign, then it is referred to as the _____ of numbers.

  1. Series

  2. Progression

  3. Arithmetic Progression

  4. Geometric Prpgression
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

When each term of a sequence is connected using a $+$ or a $-$ sign, then it is referred to as the series of numbers. For example, $2 + 5 + 8 + 11 + 15 + 18 + ....$ is a series of numbers. Thus the correct answer is '$a$'.