CAT Practice Questions: Sequences

Here is a collection of practice questions on the sequences topic collected from the past CAT question papers and other sources that I think are appropriate for preparing for the CAT exam. The entire set of practice questions in pdf format can be downloaded from this link: Sequences: Official CAT Practice Questions.

Question 1: Consider the set $S=\{2, 3, 4, \ldots , 2n+1\}$, where $n$ is a positive integer larger than 2007. Define $X$ as the average of the odd integers in $S$, and $Y$ as the average of the even integers in $S$. What is the value of $X-Y$ ? $[\textsf{Source: CAT 2007}]$

$\quad 0$

$\quad 1$

$\quad \displaystyle \frac{n}{2}$

$\quad \displaystyle \frac{n+1}{2n}$

$\quad 2008$

Video Explanation

Question 2: Consider a sequence where the $n$th term is given by $t_n = \displaystyle \frac{n}{n+2}$, $n=1, 2, 3, \ldots$. The value of $t_3 \times t_4 \times t_5 \times \ldots \times t_{53}$ equals: $[\textsf{Source: CAT 2006}]$

$\quad \displaystyle \frac{2}{495}$

$\quad \displaystyle \frac{2}{477}$

$\quad \displaystyle \frac{12}{55}$

$\quad \displaystyle \frac{1}{1485}$

$\quad \displaystyle \frac{1}{2970}$

Video Explanation

Question 3: The sum of 3rd and 15th elements of an arithmetic progression is equal to the sum of 6th, 11th and 13th elements of the same progression. Then which element of the series should necessarily be equal to zero ? $[\textsf{Source: CAT 2003}]$

$\quad $1st

$\quad $9th

$\quad $12th

$\quad $None of these

Video Explanation

Question 4: There are 8436 steel balls, each with a radius of 1 centimeter, stacked in a pile, with 1 ball on top, 3 balls in the second layer, 6 in the third layer, 10 in the fourth, and so on. The number of horizontal layers in the pile is: $[\textsf{Source: CAT 2003}]$

$\quad 34$

$\quad 38$

$\quad 36$

$\quad 32$

Video Explanation

Question 5: The infinite sum $1 + \displaystyle \frac{4}{7} + \displaystyle \frac{9}{7^2} + \displaystyle \frac{16}{7^3} + \displaystyle \frac{25}{7^4} + \ldots$ equals: $[\textsf{Source: CAT 2003 Retest}]$

$\quad \displaystyle \frac{27}{14}$

$\quad \displaystyle \frac{21}{13}$

$\quad \displaystyle \frac{49}{27}$

$\quad \displaystyle \frac{256}{147}$

Video Explanation

Question 6: If the sum of the first 11 terms of an arithmetic progression equals that of the first 19 terms, then what is the sum of the first 30 terms ? $[\textsf{Source: CAT 2004}]$

$\quad 0$

$\quad -1$

$\quad 1$

$\quad \textrm{not unique}$

Video Explanation

Question 7: Let
$$ y= \displaystyle \frac{1}{2 + \displaystyle \frac{1}{3+\displaystyle \frac{1}{2+\displaystyle \frac{1}{3+\ldots}}}} $$
What is the value of $y$ ? $[\textsf{Source: CAT 2004}]$

$\quad \displaystyle \frac{\sqrt{13}+3}{2}$

$\quad \displaystyle \frac{\sqrt{13}-3}{2}$

$\quad \displaystyle \frac{\sqrt{15}+3}{2}$

$\quad \displaystyle \frac{\sqrt{15}-3}{2}$

Video Explanation

Question 8: Consider the sequence of numbers $a_1$, $a_2$, $a_3, \ldots$ to infinity where $a_1=81.33$ and $a_2=-19$ and $a_j = a_{j-1} – a_{j-2}$ for $j \geq 3$. What is the sum of the first $6002$ terms of this sequence? $[\textsf{Source: CAT 2004}]$

$\quad -100.33$

$\quad -30$

$\quad 62.33$

$\quad 119.33$

Video Explanation

Question 9: If $a_1=1$ and $a_{n+1} – 3a_{n} + 2 = 4n$ for every positive integer $n$, then $a_{100}$ equals: $[\textsf{Source: CAT 2005}]$

$\quad 3^{99}-200$

$\quad 3^{99}+200$

$\quad 3^{100}-200$

$\quad 3^{100}+200$

Video Explanation

Question 10: Let $x=\sqrt{4 + \sqrt{4-\sqrt{4+\sqrt{4- \ldots \textrm
{to infinity}}}}}$. Then $x$ equals: $[\textsf{Source: CAT 2005}]$

$\quad 3$

$\quad \displaystyle \frac{\sqrt{13}-1}{2}$

$\quad \displaystyle \frac{\sqrt{13}+1}{2}$

$\quad \sqrt{13}$

Video Explanation

Question 11: Find the sum
$$\sqrt{1 + \displaystyle \frac{1}{1^2} + \displaystyle \frac{1}{2^2}} + \sqrt{1 + \displaystyle \frac{1}{2^2} + \displaystyle \frac{1}{3^2}} + \ldots + \sqrt{1 + \displaystyle \frac{1}{2007^2} + \displaystyle \frac{1}{2008^2}}
$$ $[\textsf{Source: CAT 2008}]$

$\quad 2008 \: – \: \displaystyle \frac{1}{2008}$

$\quad 2007 \: – \: \displaystyle \frac{1}{2007}$

$\quad 2007 \: – \: \displaystyle \frac{1}{2008}$

$\quad 2008 \: – \: \displaystyle \frac{1}{2007}$

$\quad 2008 \: – \: \displaystyle \frac{1}{2009}$

Video Explanation

Question 12: The number of common terms in the two sequences $17, 21, 25, \ldots, 417$ and $16, 21, 26, \ldots , 466$ is $[\textsf{Source: CAT 2008}]$

$\quad 78$

$\quad 19$

$\quad 20$

$\quad 77$

$\quad 22$

Video Explanation

Question 13: What is the sum of the following series:
$$\displaystyle \frac{1}{1 \times 2} + \displaystyle \frac{1}{2 \times 3} + \displaystyle \frac{1}{3 \times 4} + \ldots + \displaystyle \frac{1}{100 \times 101}$$ $[\textsf{Source: CAT 1990}]$

$\quad \displaystyle \frac{99}{100}$

$\quad \displaystyle \frac{1}{100}$

$\quad \displaystyle \frac{100}{101}$

$\quad \displaystyle \frac{101}{102}$

Video Explanation

Question 14: $N$ the set of natural numbers is partitioned into subsets $S_1=(1)$, $S_2=(2,3)$, $S_3=(4,5,6)$, $S_4=(7,8,9,10)$ and so on. The sum of the elements of the subset $S_{50}$ is $[\textsf{Source: CAT 1990}]$

$\quad 61250$

$\quad 65525$

$\quad 42455$

$\quad 62525$

Video Explanation

Question 15: A square is drawn by joining the midpoints of the sides of a given square. A third square is drawn inside the second square in the same way and this process is continued indefinitely. If a side of the first square is 8 cm, the sum of the areas of all the squares such formed (in sq. cm) is $[\textsf{Source: CAT 1990}]$

$\quad 128$

$\quad 120$

$\quad 96$

$\quad $None of these

Video Explanation

Question 16: Let $u_{n+1}=2u_{n} + 1$ where $n=(0, 1, 2, \ldots)$ and $u_0=0$, then $u_{10}$ is equal to $[\textsf{Source: CAT 1993}]$

$\quad 1023$

$\quad 2047$

$\quad 4095$

$\quad 8195$

Video Explanation

Question 17: What is the value of the following expression?
$$\displaystyle \frac{1}{2^2-1} + \displaystyle \frac{1}{4^2-1} + \displaystyle \frac{1}{6^2-1} + \ldots + \displaystyle \frac{1}{20^2-1} $$ $[\textsf{Source: CAT 2000}]$

$\quad \displaystyle \frac{9}{19}$

$\quad \displaystyle \frac{10}{19}$

$\quad \displaystyle \frac{10}{21}$

$\quad \displaystyle \frac{11}{21}$

Video Explanation

Question 18: If $a_1=1$ and $a_{n+1} = 2 a_{n} + 5$, $n=1,2,\ldots, $ then $a_{100}$ is equal to $[\textsf{Source: CAT 2000}]$

$\quad 5\times 2^{99} – 6$

$\quad 5\times 2^{99} + 6$

$\quad 6 \times 2^{99} + 5$

$\quad 6 \times 2^{99} – 5$

Video Explanation

Question 19: All the page numbers from a book are added, beginning at page $1$. However, one page number was mistakenly added twice. The sum obtained was $1000$. Which page number was added twice? $[\textsf{Source: CAT 2001}]$

$\quad 44$

$\quad 45$

$\quad 10$

$\quad 12$

Video Explanation

Question 20: A set of consecutive positive integers beginning with $1$ is written on the blackboard. A student came along and erased one number. The average of the remaining numbers is $35\frac{7}{17}$. What was the number erased? $[\textsf{Source: CAT 2001 and 1982 AMC}]$

$\quad 7$

$\quad 8$

$\quad 9$

$\quad $None of these

Video Explanation

Question 21: A student finds the sum $1+2+3+\ldots$ as his patience runs out. He found the sum as $575$. When the teacher declared the result wrong, the student realized that he missed a number. What was the number the student missed? $[\textsf{Source: CAT 2002}]$

$\quad 16$

$\quad 18$

$\quad 14$

$\quad 20$

Video Explanation

Question 22: If $X_{n} = (-1)^n X_{n-1}$ and $X_0=a$, then $[\textsf{Source: CAT 2002}]$

$\quad X_n$ is positive for $n=$ even

$\quad X_n$ is negative for $n=$ even

$\quad X_n$ is positive for $n=$ odd

$\quad $None of these

Video Explanation

Question 23: Let $S=2x+5x^2+9x^3+14x^4+20x^5+ \ldots$ to infinity, and $-1 \lt x \lt 1$. The coefficient of $n^{th}$ term is $\displaystyle \frac{n(n+3)}{3}$. The sum $S$ is $[\textsf{Source: CAT 2002}]$

$\quad \displaystyle \frac{x(2-x)}{(1-x)^3}$

$\quad \displaystyle \frac{(2-x)}{(1-x)^3}$

$\quad \displaystyle \frac{x(2-x)}{(1-x)^2}$

$\quad $None of these

Video Explanation

Question 24: The $288^{\textrm{th}}$ term of the series $a, b, b, c, c, c, d, d, d, d, e, e, e, e, e, f, f, f, f, f, f \ldots$ is $[\textsf{Source: CAT 2003}]$

$\quad u$

$\quad v$

$\quad w$

$\quad x$

Video Explanation

Question 25: The fourth term of an arithmetic progression is $8$. What is the sum of the first $7$ terms of the arithmetic progression? $[\textsf{Source: CAT 1994}]$

$\quad 7$

$\quad 64$

$\quad 56$

$\quad $cannot be determined

Video Explanation

Question 26: A group of $630$ children is arranged in rows for a group photograph session. Each row contains three fewer children than the row in front of it. What number of rows is not possible? $[\textsf{Source: CAT 2006}]$

$\quad 3$

$\quad 4$

$\quad 5$

$\quad 6$

$\quad 7$

Video Explanation

Question 27: A young girl counted in the following way on the fingers of her left hand. She started calling the thumb 1, the index finger 2, middle finger 3, ring finger 4, little finger 5, then reversed direction, calling the ring finger 6, middle finger 7, index finger 8, thumb 9 and then back to the index finger for 10, middle finger for 11 and so on. She counted up to 1994. She ended on her $[\textsf{Source: CAT 2015}]$

$\quad $thumb

$\quad $index finger

$\quad $middle finger

$\quad $ring finger

Video Explanation

Question 28: The integers $1, 2, 3, \ldots, 40$ are written on a blackboard. The following operation is then repeated $39$ times. In each repetition, any two numbers, say $a$ and $b$, currently on the blackboard are erased and a new number $a+b-1$ is written. What will be the number left on the board at the end ? $[\textsf{Source: CAT 2008}]$

$\quad 820$

$\quad 821$

$\quad 781$

$\quad 819$

$\quad 780$

Video Explanation

Question 29: The price of Darjeeling tea (in rupees per kilogram) is $100+0.10n$, on the $n$th day of 2007 ($n=1, 2, \ldots, 100)$), and then remains constant. On the other hand, the price of Ooty tea (in rupees per kilogram) is $89+0.15n$, on the $n$th day of 2007 ($n=1, 2, \ldots, 365)$. On which date in 2007 will the prices of these two varieties of tea be equal? $[\textsf{Source: CAT 2007}]$

$\quad $April 11

$\quad $May 20

$\quad $April 10

$\quad $June 30

$\quad $May 21

Video Explanation

Question 30: Let $D$ be a recurring decimal of the form $D=0.a_1a_2a_1a_2a_1a_2\ldots,$ where digits $a_1$ and $a_2$ lie between $0$ and $9$. Further, at most one of them is zero. Which of the following numbers necessarily produces an integer, when multiplied by $D$ ? $[\textsf{Source: CAT 2000}]$

$\quad 18$

$\quad 108$

$\quad 198$

$\quad 288$

Video Explanation

Question 31: If the harmonic mean between two positive numbers is to their geometric mean as $12:13$, then the numbers could be in the ratio $[\textsf{Source: CAT 1994}]$

$\quad 12:13$

$\quad \displaystyle \frac{1}{12}: \displaystyle \frac{1}{13}$

$\quad 4:9$

$\quad 2:3$

Video Explanation

Question 32: $ABCDEFGH$ is a regular octagon. $A$ and $E$ are opposite vertices of the octagon. A frog starts jumping from vertex to vertex, beginning from $A$. From any vertex of the octagon except $E$, it may jump to either of the two adjacent vertices. When it reaches $E$, the frog stops and stays there. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending in $E$. Then, what is the value of $a_{2n-1}$ ? $[\textsf{Source: CAT 2000}]$

$\quad 0$

$\quad 4$

$\quad 2n-1$

$\quad $None of the above

Video Explanation