Arithmetic Progression

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 $\text{361}$.

If $b^2,a^2,c^2$ are in AP then $a+b, b+c,c+a$ will be in 

Let the sets $A=\left\{2, 4, \mathrm{6,} 8\dots \right\}$ and $B=\left\{3, 6, 9,12\dots \right\}$ such that $n\left(A\right)=200$ and  $n\left(B\right)=250.$ If $n\left(A\cup B\right)=k$, then $\frac{k}{100}$ is equal to

If $a,b,c$ are in $\mathrm{A}.\mathrm{P}.$ and $A,B,C$ are in $\mathrm{G}.\mathrm{P}.$ (common ratio $\ne 1$ ), then which of the following is/are correct?

For the given sequence $10,18,26,32, \dots$ the number $5050$ is the:

The sum of $n$ terms of a series is given by $S_n=4n^2+6n$. Then the nature of the given series is :

If $a, b$ and $c$ are positive real numbers satisfying the equation $25\left(9a^2+b^2\right)+9c^2-15(5ab+bc+3ca)=0$ and $\frac{c-a}{b-a}=\lambda .$ Then $\frac{5}{\lambda }$ is

The sum of the first and third term of an arithmetic series is $12$ and the product of first and second term is $24,$ then first term is

In an ordered set of four numbers, the first $3$ are in A.P. and the last $3$ are in G.P., whose common ratio is $\frac{7}{4}.$ If the product of the first and fourth of these numbers is $49,$ then the product of the second and third of these is:

If $p, q, r, s, t$ and $u$ are in $AP,$ then difference $(t-r)$ is equal to

If the sides of the triangle ABC are in A.P. and the greatest angle is double the smallest. The ratio of the sides of triangle ABC is

If the sides of the right angled triangle are in A.P., then the sum of sines of the two acute angles is

In a $∆\mathrm{A}\mathrm{B}\mathrm{C}$ angles A,B,C are in increasing A.P. and $sin\left(B+2C\right)=\mathrm{ }-\frac{1}{2}\mathrm{ }\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{ }A\mathrm{ }=\mathrm{ }?$

Find the common difference of an $A.P.$ whose first term is $100$ and the sum of whose first six terms is five times the sum of next six terms.

In a set of four numbers, if first three terms are in G.P. and the last three terms are in A.P. $\mathrm{ }$ with common difference 6, then sum of the four numbers, when the first and the last terms are equal is?

In an arithmetic progression consisting of positive terms each term equals the sum of the next two terms. First term $=a.$ then the common difference of its progression is equals to?

A person has to count the $4500\mathrm{ }$currency notes. Let $a_0$ be the number of notes he counts in the $n^{\mathrm{t}\mathrm{h}}\mathrm{ }$minute. If $a_1=a_2=\dots =a_{10}=150,$and the terms $a_{10},a_{11},\dots$ are in $\mathrm{A}.\mathrm{P}$ with a common difference of $-2,\mathrm{ }$then the time taken by him to count all notes is (minutes)

If the $\mathrm{ }{2}^{\mathrm{n}\mathrm{d}}\mathrm{ },5^{\mathrm{t}\mathrm{h}}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }{9}^{\mathrm{t}\mathrm{h}}\mathrm{ }$ terms of a non-constant A.P are in G.P then the common ratio of G.P is

How many$3$ digit number are divisible by $6$ in all?

Let $x_1, x_2, x_3\dots ..$ be positive integers in $\mathrm{AP}$ such that $x_1+x_2+x_3=12$ and $x_4+x_5=14$ then $x_7=$