Suppose $x_{1},x_{2},x_{3},...,x_{100}$ are in arithmetic progression such that $x_{5}=-4$ and $2x_{6}+2x_{9}=x_{11}+x_{13}$, Then,$x_{100}$ equals
For any natural number $n$ let $a_{n}$ be the largest integer not exceeding $\sqrt{n}$. Then the value of $a_{1}+a_{2}+.....+a_{50}$ is
The sum of the infinite series $\cfrac{1}{5}\left(\cfrac{1}{5} - \cfrac{1}{7}\right) + \left(\cfrac{1}{5}\right)^2 \left(\left(\cfrac{1}{5}\right)^2 - \left(\cfrac{1}{7}\right)^2\right) + \left(\cfrac{1}{5}\right)^3 \left(\left(\cfrac{1}{5}\right)^3 - \left(\cfrac{1}{7}\right)^3\right) + ......$ is equal to
Consider the sequence $t_1 = 1, t_2 = -1$ and $t_n = \left(\cfrac{n - 3}{n - 1}\right)t_{n - 2}$ for $n \geq 3$. Then, the value of the sum $\cfrac{1}{t_2} + \cfrac{1}{t_4} + \cfrac{1}{t_6} + ....... +\cfrac{1}{t_{2022}} + \cfrac{1}{t_{2024}}$, is
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
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 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
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
For a real number x , if $\frac{1}{2}, \frac{\log _3\left(2^x-9\right)}{\log _3 4}$, and $\frac{\log _5\left(2^x+\frac{17}{2}\right)}{\log _5 4}$ are in an arithmetic progression, then the common difference is
The value of $1+\left(1+\frac{1}{3}\right) \frac{1}{4}+\left(1+\frac{1}{3}+\frac{1}{9}\right) \frac{1}{16}+\left(1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}\right) \frac{1}{64}+\cdots$ is
Let $a_n=46+8 n$ and $b_n=98+4 n$ be two sequences for natural numbers n ≤ 100. Then, the sum of all terms common to both the sequences is
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
On day one, there are 100 particles in a laboratory experiment. On day n, where n ≥ 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 a non-decreasing sequence of N numbers $a_1, a_2, \ldots, a_N$ 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
The average of all 3-digit terms in the arithmetic progression 38, 55, 72, ..., is
In the set of consecutive odd numbers $\{1, 3, 5, \dots, 57\}$, there is a number $k$ such that the sum of all the numbers less than $k$ is equal to the sum of all the numbers greater than $k$. Then, $k$ equals:
Let $a_{n}$ be the $n^{th}$ 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
The average number of copies of a book sold per day by a shopkeeper is 60 in the initial seven days and 63 in the initial eight days, after the book launch. On the ninth day, she sells 11 copies less than the eighth day, and the average number of copies sold per day from second day to ninth day becomes 66. The number of copies sold on the first day of the book launch 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
