Arithmetic Progressions

For each positive integer $\text{k}$, let ${\text{S}}_{\text{k}}$ denote the increasing arithmetic sequence of integers whose first term is $\text{1}$ and whose common difference is $\text{k}$. For example,${\text{ S}}_{\text{3}}$ is the sequence $\text{1, 4, 7, 10 ..............}$. Find the number of values of $\text{k}$ for which ${\text{S}}_{\text{k}}$ contain the term $\text{361}$.

In a triangle $∆ABC$, if $\tan (A/2),\tan (B/2),\tan (C/2)$ are in Arithmetic progression, then which of the following option is always correct?

The arithmetic mean of the following discrete data $12, 14, 20, 23, 25, 32$ is given by

The arithmetic mean of five natural numbers is $40 .$ The largest exceeds the smallest number by $10 .$ If $\alpha$ is the maximum possible value for the largest of these $5$ numbers, then the number of positive integral divisors of $\alpha$ is ______

The mean marks of $25$ boys in a class is $61$ and the mean marks of $35$ girls in the same class is $58.$ Then the mean of all $60$ students is

The interior angles of a polygon are all obtuse and are in A.P. If the smallest angle is ${120}^{°}$ and common difference of this A.P. is $5^{°},$ then the number of sides of the polygon is___________

If ${\log }_{3}2, {\log }_3\left(2^x-5\right)$ and ${\log }_3\left(2^x-\frac{7}{2}\right)$ are in Arithmetic Progression, then the value of $x$ is _________.

${\log }_{10}2, {\log }_{10}\left(2^x-1\right)$ and ${\log }_{10}\left(2^x+3\right)$ are three consecutive terms of an $\mathrm{AP}$ for

If $a_1, a_2,a_3\dots .$ are terms of $AP$ such that $a_1+a_5+a_{10}+a_{15}+a_{20}+a_{24}=225,$ then the sum of first $24$ terms is

Let ${\mathrm{a}}_1,\mathrm{ }{\mathrm{a}}_2,\mathrm{ }{\mathrm{a}}_3,\mathrm{ }\dots \dots$ be an A.P. with ${\mathrm{a}}_6=2$ . Then the common difference of this $\mathrm{A}.\mathrm{P}.$ , which maximize the product ${\mathrm{a}}_1\mathrm{ }{\mathrm{a}}_4\mathrm{ }{\mathrm{a}}_5,\mathrm{ }$ is :

The arithmetic mean of first $n$ odd natural number is

If $\log 2, \log \left(2^n-1\right)$ and $\log \left(2^n+3\right)$ are in arithmetic progression, then the value of $n$ will be

If ${\sum }_{r=1}^n\left(2r+1\right)=440$ , then $n=$…..

The coefficient of $x^{49}$ in the product

$\left(x-1\right)\left(x-3\right)\mathrm{...}\left(x-99\right)$  is

If in an A.P, $S_n=q{n}^2$ and $S_m=q{m}^2$, where $S_r$ denotes the sum of $r$ terms of the A.P, then $S_q$ equals

The first three terms of a geometric sequence are $\mathrm{x},\mathrm{ }\mathrm{y},\mathrm{ }\mathrm{z}$ and these have the sum equal to 42. If the middle term $\mathrm{y}$ is multiplied by $5/4$ , the numbers $\mathrm{x},\mathrm{ }\frac{5\mathrm{y}}{4},\mathrm{ }\mathrm{z}$ now form an arithmetic sequence. The largest possible value of $\mathrm{x}$ , is-

If $\mathrm{A}.\mathrm{M}.$ between $p^{\mathrm{th}}$ and $q^{\mathrm{th}}$ terms of an $\mathrm{A}.\mathrm{P}.$ be equal to the $\mathrm{A}.\mathrm{M}.$ between $r^{\mathrm{th}}$ and $s^{\mathrm{th}}$ terms of the $\mathrm{A}.\mathrm{P}.$, then $p+q$ is equal to

The sets $S_1,S_2,S_3\dots$ are given by $S_1=\left\{\frac{2}{1}\right\}, S_2=\left\{\frac{3}{2},\frac{5}{2}\right\}, S_3=\left\{\frac{4}{3},\frac{7}{3},\frac{10}{3}\right\}, S_4=\left\{\frac{5}{4},\frac{9}{4},\frac{13}{4},\frac{17}{4}\right\},\dots$ Then, the sum of the numbers in the set $S_{25}$ is

If $a_1, a_2,a_3, ......a_{24}$ are in A.P. such that $a_1+a_5+a_{10} +a_{15}+a_{20}+a_{24} = 225$. Then $a_1+a_2+......+a_{24}$ is equal to