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:

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

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

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

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

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 $n \geq 2$ , 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

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

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

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

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

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]

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

Let $t_1, t_2, \dots$ be real numbers such that $t_1 + t_2 + \dots + t_n = 2n^2 + 9n + 13$, for every positive integer $n \geq 2$. If $t_k = 103$, then $k$ equals (TITA).

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

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