Insert $n$ arithmetic means between $x$ and $y$.

Important Questions on Arithmetic Progression

HARD

Mathematics>Algebra>Arithmetic Progression>Arithmetic Mean

Let $S_n=\underset{k=1}{\overset{4n}{\Sigma }}{\left(-1\right)}^{\frac{k\left(k+1\right)}{2}}{k}^2$. Then $S_n$ can take value(s)

HARD

Mathematics>Algebra>Arithmetic Progression>Arithmetic Mean

The houses on one side of a road are numbered using consecutive even numbers. The sum of the numbers of all the houses in that row is $170$. If there are at least $6$ houses in that row and $a$ is the number of the sixth house, then

EASY

Mathematics>Algebra>Arithmetic Progression>Arithmetic Mean

In an arithmetic progression, if the $k^{th}$ term is $5k+1$$,$ then the sum of first $100$ terms is

HARD

Mathematics>Algebra>Arithmetic Progression>Arithmetic Mean

The number of terms in an $A.P.$ is even, the sum of the odd terms in it is $24$ and that the even terms is $30.$ If the last term exceeds the first term by $10\frac{1}{2},$ then the number of terms in the $A.P.$ is 

MEDIUM

Mathematics>Algebra>Arithmetic Progression>Arithmetic Mean

If $x, y, z$ are positive numbers in $\mathrm{A}.\mathrm{P}.$ and ${\tan }^{-1}x$, ${\tan }^{-1}y$ and  ${\tan }^{-1}z$ are also in $\mathrm{A}.\mathrm{P}.$, then which of the following is correct.

MEDIUM

Mathematics>Algebra>Arithmetic Progression>Arithmetic Mean

If $a^{1/x}=b^{1/y}=c^{1/z}$ and $a,b,c$ are in geometrical progression, then $x,y,z$ are in

HARD

Mathematics>Algebra>Arithmetic Progression>Arithmetic Mean

Let $a, b, c, d, e$ be natural numbers in an arithmetic progression such that $a+b+c+d+e$ is the cube of an integer and $b+c+d$ is a square of an integer. The least possible value of the number of digits of $c$ is

MEDIUM

Mathematics>Algebra>Arithmetic Progression>Arithmetic Mean

The number of terms common to the two A.P.’s $3,7,11,\dots ,407$ and $2,9,16,\dots ,709$ is ____________.

HARD

Mathematics>Algebra>Arithmetic Progression>Arithmetic Mean

Let the sum of the first three terms of an A.P. be $39$ and the sum of its last four terms be $178$. If the first term of this A.P. is $10,$ then the median of the A.P. is :

MEDIUM

Mathematics>Algebra>Arithmetic Progression>Arithmetic Mean

Three positive numbers form an increasing $G.P.$ If the middle term in this $G.P.$ is doubled, the new numbers are in $A.P.$ Then the common ratio of the $G.P.$ is :

HARD

Mathematics>Algebra>Arithmetic Progression>Arithmetic Mean

Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than $200$ and less than $220.$  If the second term in it is $12$, then its $4^{\mathrm{th}}$ term is :

HARD

Mathematics>Algebra>Arithmetic Progression>Arithmetic Mean

If ${}^n{C}_4,{ }^n{C}_5$ and ${}^n{C}_6$ are in A.P., then $n$ can be

MEDIUM

Mathematics>Algebra>Arithmetic Progression>Arithmetic Mean

Let $a_1, a_2, a_3,\dots \dots ,a_{49}$ be in $A.P.$ such that $\underset{k=0}{\overset{12}{\Sigma }}{a}_{4k+1}=416$ and $a_9+a_{43}=66.$ If $a_1^2+a_2^2+\dots +a_{17}^2=140m,$ then $m$ is equal to:

MEDIUM

Mathematics>Algebra>Arithmetic Progression>Arithmetic Mean

If the sum of the first $n$ terms of the series $\sqrt{3}+ \sqrt{75}+ \sqrt{243}+ \sqrt{507}+\dots$ is $435\sqrt{3},$ then $n$ equals:

MEDIUM

Mathematics>Algebra>Arithmetic Progression>Arithmetic Mean

Let $a_1, a_2, a_3,\dots a_n,\dots$ ,be in A.P. If $a_3+a_7+a_{11}+a_{15}=72$, then the sum of its first $17$ terms is equal to :

MEDIUM

Mathematics>Algebra>Arithmetic Progression>Arithmetic Mean

If the sum and product of the first three terms in an $A.P.$ are $33$ and $1155$, respectively, then a value of its ${11}^{\mathrm{t}\mathrm{h}}$ term is:

MEDIUM

Mathematics>Algebra>Arithmetic Progression>Arithmetic Mean

Let the sum of the first $n$ terms of a non-constant $A.P., a_1, a_2, a_3, ...., a_n$ be $50n+\frac{n(n-7)}{2}A,$ where $A$ is a constant. If $d$ is the common difference of this $A.P.$, then the ordered pair $\left(d, a_{50}\right)$ is equal to

HARD

Mathematics>Algebra>Arithmetic Progression>Arithmetic Mean

For any three positive real numbers $a, b$ and $c$. If $9\left(25a^2+b^2\right)+25\left(c^2-3ac\right)=15b\left(3a+c\right).$ Then

HARD

Mathematics>Algebra>Arithmetic Progression>Arithmetic Mean

Let $X$ be the set consisting of the first $2018$ terms of the arithmetic progression $\mathrm{1,6},11,\dots ,$ and $Y$ be the set consisting of the first $2018$ terms of the arithmetic progression $\mathrm{9,16,23},\dots$ . Then, the number of elements in the set $X\cup Y$ is___.

MEDIUM

Mathematics>Algebra>Arithmetic Progression>Arithmetic Mean

The sum of all two digit positive numbers which when divided by $7$ yield $2$ or $5$ as remainder is