Progression CAT PYQs , 56 Questions with Solutions

In the set of consecutive odd numbers $\{1, 3, 5, \ldots, 57\}$, there is a number $k$ such that the sum of all elements less than $k$ is equal to the sum of all elements greater than $k$. Then, $k$ equals

Let $a _ { n }$ be the $n ^ { \text {th } }$ term of a decreasing infinite geometric progression. If $a _ { 1 } + a _ { 2 } + a _ { 3 } = 52$ and $a _ { 1 } a _ { 2 } + a _ { 2 } a _ { 3 } + a _ { 3 } a _ { 1 } = 624$, then the sum of this geometric progression is

In an arithmetic progression, if the sum of fourth, seventh and tenth terms is 99, and the sum of the first fourteen terms is 497, then the sum of first five terms is

Suppose $x_1, x_2, x_3, \ldots, x_{100}$ are in arithmetic progression such that $x_5 = -4$ and $2 \times x_6 + 2 \times x_9 = x_{11} + x_{13}$. Then, $x_{100}$ equals

The sum of the infinite series $\frac{1}{5}\left(\frac{1}{5}-\frac{1}{7}\right)+\left(\frac{1}{5}\right)^2\left(\left(\frac{1}{5}\right)^2-\left(\frac{1}{7}\right)^2\right)+\left(\frac{1}{5}\right)^3\left(\left(\frac{1}{5}\right)^3-\left(\frac{1}{7}\right)^3\right)+\cdots$ is equal to

Consider the sequence $t_1 = 1$, $t_2 = -1$, and

Then, the value of the sum

is:

Based on the given pattern find the next term of the given series: 5, 6, 16, 57, 244, 1245, ?

In a school, the number of students in each class, from Class I to X, in that order, are in an arithmetic progression. The total number of students from Class I to V is twice the total number of students from Class VI to X. If the total number of students from Class I to IV is 462, how many students are there in Class VI?

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

Separately, Jack and Sristi invested the same amount of money in a stock market. Jack’s invested amount kept getting reduced by $50\%$ every month. Sristi’s investment also reduced every month, but in an arithmetic progression with a common difference of Rs. $15000$. They both withdrew their respective amounts at the end of the sixth month. They observed that if they had withdrawn their respective amounts at the end of the fourth month, the ratio of their amounts would have been the same as the ratio after the sixth month. What amount of money was invested by Jack in the stock market?

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 $n^{\text {th }}$ day exceeds one million, then the lowest possible value of $n$ 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 both the series $a_1, a_2, a_3, \ldots$ and $b_1, b_2, b_3 \ldots$ 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

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

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

Consider the sequence $a_{n+1} = \dfrac{1}{1 + \dfrac{1}{a_n}}$ for $n = 1, 2, \ldots, 2008, 2009$ where $a_1 = 1$. Find the value of $a_1a_2 + a_2a_3 + a_3a_4 + \ldots + a_{2008}a_{2009}$.

There are three sections in a question paper and each section has 10 questions. First section only has multiple-choice questions, and 2 marks will be awarded for each correct answer. For each wrong answer, $0.5$ marks will be deducted. Any unattempted question in this section will be treated as a wrong answer. Each question in the second section carries 3 marks, whereas each question in the third section carries 5 marks. For any wrong answer or un-attempted question in the second and third sections, no marks will be deducted. A student’s score is the addition of marks obtained in all the three sections. What is the sixth highest possible score?

If in a decreasing AP, sum of all the terms except the first term is −75 while the sum of all the terms is 0. If the difference between the 9th term and the 13th term is 20, the find the first term of the AP?

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

A marble is dropped from a height of 3 metres onto the ground. After hitting the ground, it bounces and reaches 80% of the height from which it was dropped. This repeats multiple times. Each time it bounces, the marble reaches 80% of the height previously reached. Eventually, the marble comes to rest on the ground. What is the maximum distance that the marble travels from the time it was dropped until it comes to rest?

If both the sequences $x, a_1, a_2, y$ and $x, b_1, b_2, z$ are in A.P. and it is given that $y > z$ and $z < x$, then which of the following values can $\frac{(a_1 - a_2)}{(b_1 - b_2)}$ possibly take?

Consider the arithmetic progression $3,7,11, \ldots$ 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

On day one, there are 100 particles in a laboratory experiment. On day $n$ , where $n \geq 2$ , 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

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

Given below are two statements Statement I: In the sequence of numbers 30, 90, 182, 306, 462, P….. the term P is 650. Statement II: There are 8 digits in 9⁸ when it is expressed in decimal form. In light of the above statements, choose the correct answer from the options given below

Natural numbers are divided into groups as follows:

• Group 1: (1)
• Group 2: (2, 3, 4)
• Group 3: (5, 6, 7, 8, 9)
• Group 4: (10 … 16)
  …and so on, where the k‑th group contains (2k–1) numbers.
  What is the sum of the numbers in the 15th group?

For a sequence of real numbers $x_1, x_2, \ldots, x_n$, if $x_1 - x_2 + x_3 - \cdots + (-1)^{\,n+1} x_n = n^2 + 2n$ for all natural numbers $n$, then the sum $x_{49} + x_{50}$ equals

Find the sum of the following series (with infinite terms): 2√2, 4/√3, 4√2/3, ……

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

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

Let the m-th and n-th terms of a Geometric progression be $\frac{3}{4}$ and 12, respectively, when m < n. If the common ratio of the progression is an integer r, then the smallest possible value of r + n - m is

Find the missing number in the series:

1680, 1168, 1119, 903, ?, 814

Find the missing number in the series:

110, 55, 55, 110, ?, 3520

Find the missing number in the series:

8, 5, 7, 18, 80, ?

Find the missing number in the series:

11, 12, 17, 30, 59, ?

Find the missing number in the series:

?, 237, 366, 473, 547, 577

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

If $a_1 + a_2 + a_3 + \dots + a_n = 3(2^{n+1} - 2),$ then find the value of $a_{11}.$

If $a_1, a_2, \ldots$ are in Arithmetic Progression (A.P.), then the value of the expression

$\dfrac{1}{\sqrt{a_1} + \sqrt{a_2}} + \dfrac{1}{\sqrt{a_2} + \sqrt{a_3}} + \cdots + \dfrac{1}{\sqrt{a_n} + \sqrt{a_{n+1}}}$
is equal to:

The number of common terms in the two sequences: 15, 19, 23, 27, ...... , 415 and 14, 19, 24, 29, ...... , 464 is

If (2n+1) + (2n+3) + (2n+5) + ... + (2n+47) = 5280 , then what is the value of 1+2+3+ ... +n ? [TITA]

A new species lays exactly 120 eggs out of which 50% are male and 50% are female. The female insect hatch and grow in a span of 20 days to lay eggs by themselves. On 1st April, 2018, an insect laid 120 eggs. Find how many eggs will be hatched, (approx) by the end of may 2018.

Amar consumed 100 laddoos from Monday to Friday. Each day, he consumed 6 more laddoos than the previous day. How many laddoos did he consume on Wednesday?

When opening his fruit shop for the day a shopkeeper found that his stock of apples could be perfectly arranged in a complete triangular array: that is, every row with one apple more than the row immediately above, going all the way up ending with a single apple at the top.

During any sales transaction, apples are always picked from the uppermost row, and going below only when that row is exhausted.

When one customer walked in the middle of the day she found an incomplete array in display having 126 apples totally. How many rows of apples (complete and incomplete) were seen by this customer? (Assume that the initial stock did not exceed 150 apples.)

Let x, y, z be three positive real numbers in a geometric progression such that x < y < z. If 5x, 16y, and 12z are in an arithmetic progression then the common ratio of the geometric progression is

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

Simplify the given expression: 1 + 1/2 + 1/(2²) + 1/(2³) + ... + 1/(2^(n-1)) = ?

David has an interesting habit of spending money. He spends exactly £$X$ on the $X$th day of a month. For example, he spends exactly £$5$ on the $5$th of any month. On a few days in a year, David noticed that his cumulative spending during the last $4$ consecutive days can be expressed as $2^N$ where $N$ is a natural number. What can be the possible value(s) of $N$?

An antique store has a collection of $8$ clocks. At a particular moment, the displayed times on $7$ of the $8$ clocks were as follows: $1:55$ pm, $2:03$ pm, $2:11$ pm, $2:24$ pm, $2:45$ pm, $3:19$ pm and $4:14$ pm. If the displayed times of all $8$ clocks form a mathematical series, then what was the displayed time on the remaining clock?

Let $a_1, a_2, \ldots, a_{3n}$ be an arithmetic progression with $a_1 = 3$ and $a_2 = 7$. If $a_1 + a_2 + \ldots + a_{3n} = 1830$, then what is the smallest positive integer $m$ such that $m(a_1 + a_2 + \ldots + a_n) > 1830$?

If the square of the $7^{\text{th}}$ term of an arithmetic progression with positive common difference equals the product of the $3^{\text{rd}}$ and $17^{\text{th}}$ terms, then the ratio of the first term to the common difference is:

An infinite geometric progression a₁, a₂, a₃,… has the property that aₙ = 3(aₙ₊₁ + aₙ₊₂ + …) for every n ≥ 1. If the sum a₁ + a₂ + a₃ + … = 32, then a₅ is

If a 1 = $\frac{1}{2 × 5}$ , a 2 = $\frac{1}{5 × 8}$ , a 3 = $\frac{1}{8 × 11}$ ,...., then a 1 + a 2 + a 3 + ...... + a 100 is

The sum of series, $(-100) + (-95) + (-90) + \cdots + 110 + 115 + 120$, is:

Find out next number in the series 2, 7, 17, 34, 60, ......