Arithmetic Progressions

How many three-digit numbers are divisible by $6$ in all?

How many natural numbers are there between $23$ and $100$ which are exactly divisible by $6$?

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 the $15\mathrm{th}$ term of an arithmetic series is $143$ and the $31\mathrm{st}$ term is $183$, then find the sum of first $13$ terms.

If the $15\mathrm{th}$ term of an arithmetic series is $143$ and the $31\mathrm{st}$ term is $183$, then find the value of $100th$ term

If the $15\mathrm{th}$ term of an arithmetic series is $143$ and the $31\mathrm{st}$ term is $183$, then find the value of $5th$ term

If the $15\mathrm{th}$ term of an arithmetic series is $143$ and the $31\mathrm{st}$ term is $183$, then find the value of first term

The number of positive integers $x$ which satisfy the condition $\left[\frac{x}{99}\right]=\left[\frac{x}{101}\right]$, where $\left[·\right]$ is greatest integer functions

The number of $5$-tuples $\left(a,b,c,d,e\right)$ of positive integers such that

I. $a,b,c,d,e$ are the measures of angles of a convex pentagon in degrees

II. $a\le b\le c\le d\le e$

III. $a,b,c,d,e$ are in arithmetic progression, is

The sum of $n$ consecutive terms of an arithmetic progression consisting of integers is $161$. Then, a possible value of $n$ is

What is the ${16}^{\mathrm{th}}$ term of this A.P series $11, 9, 7, 5, 3.....?$

what is the ${18}^{th}$term of this A,P. series $2,7,12,17......?$

What is  the  ${16}^{th}$ term of the A.P. $5,9,13,17,21,25,.........?$

The first four terms of an arithmetic sequence are $p, 9, \left(3p-q\right)$ and $\left(3p+q\right)$. What is the sum of digits of the ${2010}^{\mathrm{th}}$ term of the sequence?

A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have $3$ rows of small congruent equilateral triangles, with $5$ small triangles in the base row. Total $N$ number of toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of $2003$ small equilateral triangles. Then find the sum of all digits of $N.$

A circle with area $A_1$ is contained in the interior of a larger circle with area $A_1+A_2$. If the radius of the larger circle is $3$, and if $A_1, A_2, A_1+A_2$ is an arithmetic progression, then the radius of the smaller circle is 

If the sum of all five numbers which are in Arithmetic Progression is $115$ and the product of the second and fourth term is $493$. Find the fifth term of the Arithmetic Progression.

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 _________.