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 $\text{361}$.

Let $a_n$ be $n^{\mathrm{th}}$ term of the series $5+8+14+23+35+50+.......$and $S_n=\sum _{k=1}^n{a}_k$. Then $S_{30}-a_{40}$ is equal to 

The number of elements in the set $\{n\in \mathrm{\mathbb{N}}:10\le n\le 100$ and $3^n-3$ is a multiple of $7\}$ is _______.

The sum of all those terms, of the arithmetic progression $3, 8, 13, ..., 373$, which are not divisible by $3$, is equal to ________.

If $gcd\left(m, n\right)=1$ and $1^2-2^2+3^2-4^2+....+{\left(2021\right)}^2-{\left(2022\right)}^2+{\left(2023\right)}^2=1012m^{2}n$ then $m^2-n^2$ is equal to

If $n\in \left[10,100\right]$ and $n\in N$, then how many such $n$ are possible where $3^n-3$ is divisible by $7$.

If $3,8,13,.....373$ is in AP then sum of terms which is not divisible by $3$ is

If $1^2-2^2+3^2-4^2+....-{\left(2022\right)}^2+{\left(2023\right)}^2=m^{2}n$ where $m,n\in N$ and $m>19$ then $n-m^2$ is

Let $a_1, a_2, ..., a_n$ are in arithmetic progression having common difference $d$, then the value of $\underset{n\to \infty }{\lim }\sqrt{\frac{d}{n}}\left(\frac{1}{\sqrt{a_1}+\sqrt{a_2}}+\frac{{\displaystyle 1}}{{\displaystyle \sqrt{a_2}+\sqrt{a_3}}}+.....+\frac{{\displaystyle 1}}{{\displaystyle \sqrt{a_{n-1}}+\sqrt{a_n}}}\right)$ is

Sum of first $20$ terms of the series $5, 11, 19, 29, 41 .............$ is

Three non-zero numbers $a, b$ and $c$ are in A.P. Increasing $a$ by $1$ or increasing $c$ by $2$, the numbers are G.P., then find $b$.

In an Arithmetic Progression, sixth term is one more than twice the third term. The sum of the fourth and fifth terms is five times the second term. Find the tenth term of the Arithmetic Progression.

If $\frac{1}{p+q}, \frac{1}{q+r} \mathrm{and} \frac{1}{r+p}$ are in AP, then 

Find the three numbers $a,b,c$ between $2$ and $18$ such that

(i) their sum is $25$

(ii) the numbers $2,a,b$ are consecutive terms of an AP

(iii) the numbers $b,c,18$ are consecutive terms of GP

Let $a_1,a_2,a_3\dots ,a_n$ be in A.P. and $a_3,a_5,a_8,b_1,b_2,b_3\dots ,b_n$ be in G.P. If $a_9=40$, then the value of $\sum _{i=1}^2{a}_i^2$ is

Six numbers are in A.P. such that their sum is $3$. The first term is $4$ times the third term. Then the fifth term is

Find the summation upto $n^{th}$ term of the series $1^2-2^2+3^2-4^2+5^2.............$

Let be an arithmetic sequence such that arithmetic mean of $a_1,a_3,a_5\dots \dots \dots \dots \dots .a_{97},a_{99}$ is $1$. Then the value of $\left|\sum _{r=1}^{50}{\left(-1\right)}^{\frac{r(r+1)}{2}}·a_{2r-1}\right|=$

If $a,b,c$ are in A.P. and the equations $\left(b-c\right)x^2+\left(c-a\right)x+a-b=0$ and $2\left(c+a\right)x^2+\left(b+c\right)x=0$ have a common root then