Arithmetic Progression

For what value of $n$, $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ is the arithmatic mean of $a$ and $b$.

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 an $AP$, the first term is $x$ and the sum of the first $n$ terms is zero. What is the sum of next $m$ terms?

The first and the second terms of an $AP$ are $\frac{5}{2}$ and $\frac{23}{12}$ respectively. If $n^{\mathrm{th}}$ term is the largest negative term, what is the value of $n$?

$p, q, r$ and $s$ are in $AP$ such that $p+s=8$ and $qr=15$. What is the difference between largest and smallest numbers?

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 

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

The sum of $12 \mathrm{A}.\mathrm{M}.$'s between two distinct real numbers is $156$. Then the single A.M. between those two numbers is

If the $1\mathrm{st}$ term of an A.P. is $3$, the last term is $39$ and the sum of the terms is $525$, then its common difference is

If ${\mathrm{a}}_{1,}{\mathrm{a}}_2,{\mathrm{a}}_3,...,{\mathrm{a}}_{\mathrm{n}}$ are in A.P. with common difference d, then

$\mathrm{sind}\left[{\mathrm{coseca}}_1{\mathrm{coseca}}_2+{\mathrm{coseca}}_2{\mathrm{coseca}}_3+...+{\mathrm{coseca}}_{\mathrm{n}-1}{\mathrm{coseca}}_{\mathrm{n}}\right]=$

If the $5\mathrm{th}$ term of an A.P. is $\frac{4+{\tan }^2\mathrm{\alpha }}{1+{\tan }^2\mathrm{\alpha }}$ and the common difference is $\cos 2\mathrm{\alpha }$, then its $1\mathrm{st}$ term is

${\mathrm{A}}_1,{\mathrm{A}}_2,{\mathrm{A}}_3$ are three A.P.'s such that the first term of both ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$ is $3$; and ${\mathrm{A}}_2$ and ${\mathrm{A}}_3$ bear the same common difference. Moreover, the $4\mathrm{th}$ term of ${\mathrm{A}}_1$ is same as the$8\mathrm{th}$ term of ${\mathrm{A}}_2$. If the $22\mathrm{nd}$ term of ${\mathrm{A}}_1$ differs numerically from $50\mathrm{th}$ term of ${\mathrm{A}}_3$ by $7$, then the $1\mathrm{st}$ term of ${\mathrm{A}}_3$ is

If $36$ is divided into $4$ parts which are in A.P., in such a way that the product of extremes is to the product of two middle parts is $2:3$, then the third part is

The $4\mathrm{th}$ term of an A.P. is same as the $6\mathrm{th}$ term of another A.P. If the two A.P.s have the same first term, then

The first term of each of two finite A.P.'s with same number of terms is $2$ and their last terms are respectively $14$ and $17$. Then

If the $\mathrm{nth}$ term of an arithmetic progression is $2\mathrm{n}-4$, then the common difference of tin progression is

If$2\mathrm{r}-3,\mathrm{r}+5,3\mathrm{r}+2$  are in A.P., then the value of $\mathrm{r}$ is

If $5$$\mathrm{th}$ and $10$$\mathrm{th}$ terms of an A.P. are respectively $11$and $21$,then the $16$$\mathrm{th}$ term of the A.P. is