For $x \geq 0$, the least value of $K$, for which $4^{1 + x} +$ $4^{1 - x},\frac{K}{2},16^{x} + 16^{- x}$ are three consecutive terms of an A.P., is equal to :

Five numbers are in A.P., whose sum is 25 and product is 2520 . If one of these five numbers is $- 1/2$, then the greatest number amongst them is

If $\log_{e}a,\log_{e}b,\log_{e}c$ are in an A.P. and\n ,\n are also in an\n A.P., then is equal to

If $n$ is a natural number such that\n $n = p_{1}^{\alpha_{1}} \cdot p_{2}^{\alpha_{2}} \cdot p_{3}^{\alpha_{3}}\ldots\ldots p_{k}^{\alpha_{k}}$\n and are distinct primes, then show that\n .

Let 𝐴 = {2,3,6,8,9,11} and 𝐵 = {1,4,5,10,15}. Let 𝑅 be a relation on 𝐴 × 𝐵 defined by (𝑎, 𝑏)𝑅(𝑐, 𝑑) if and only if 3𝑎𝑑 − 7𝑏𝑐 is an even integer. Then the relation 𝑅 is

Let $a_{1},a_{2},a_{3},\ldots$ be a sequence of positive integers in arithmetic progression with common difference 2 . Also, let be a sequence of positive integers in geometric progression with common ratio 2 . If , then the number of all possible values of , for which the equality holds for some positive integer , is

If $gcd(m,n) = 1$ and\n $1^{2} - 2^{2} + 3^{2} - 4^{2} + \ldots. + (2021)^{2}$\n , then is\n equal to

The $8^{\text{th}\text{~}}$ common term of the series $S_{1} = 3 + 7 + 11 + 15 + 19 + \ldots.$. . , is

A pack contains $n$ cards numbered from 1 to $n$. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224 . If the smaller of the numbers on the removed cards is $k$, then

The area of the region enclosed by the curve $f(x)=max\{sin⁡x,cos⁡x\},- π ≤x ≤\nπ$ and the x-axis is

Let $a_{1},a_{2},\ldots a_{10}$ be in A.P. and be in H.P. If and , then is

Let $a_{1},a_{2},a_{3},\ldots\ldots,a_{11}$ be real numbers satisfying and for . If , then the value of is equal

Let $X$ be the set consisting of the first 2018 terms of the arithmetic progression $1,6,11,\ldots$, and $Y$ be the set consisting of the first 2018 terms of the arithmetic progression . Then, the number of elements in the set is

Let $a_{1},a_{2},a_{3},\ldots\ldots$. be an A.P. If\n ,\n , then is equal to

A software company sets up $m$ number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start n of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $m$ is equal to

Number of 4 -digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11 , is equal to

Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic\n progression. If , then the value of\n is

Let $a_{1},a_{2},a_{3},\ldots$ be terms of an A.P. If\n ,\n then equals

If $S_{1},S_{2},S_{3},\ldots\ldots,S_{n}$ are the sums of infinite geometric series whose first terms are and whose common ratios are respectively, n then find the values of .

If the sum and product of the first three terms in an A.P. are 33 and 1155 , respectively, then a value of its $11^{\text{th}\text{~}}$ term is