A.M. and Useful Results in Arithmetic Progression

The sum of natural numbers upto $200$excluding those divisible by$5$ is _____.

Find the sum to $n$ terms of $(1-\frac{1}{n})+(1-\frac{2}{n})+(1-\frac{3}{n})+\dots \dots$

The sum of a series in$\mathrm{A}.\mathrm{P}.$ is $72$the first term is $17$and the common difference $-2.$ the number of terms is _____.

If $a, b, c$ are in $\mathrm{A}.\mathrm{P}.$ as well as in $\mathrm{G}.\mathrm{P}.$ then ______.

The $\mathrm{A}.\mathrm{M}.$of two positive numbers is $40$and their $\mathrm{G}.\mathrm{M}.$is $24$. The numbers are _____.

The two arithmetic means between $-6$ and $14$is _____.

Let $a,b,c$ be in $A.P.$ If $p$ is the $A.M.$ between $a$ and $b$ and $q$ is the $A.M.$between $b$ and $c$, then prove that $b$ is the $A.M.$between $p$ and $q$.

If $A$ is the $A.M.$ between $a$ and $b$ show that $\frac{A+2a}{A-b}+\frac{A+2b}{A-a}=4$.

The sum of two numbers is $\frac{13}{6}$. An even number of arithmetic means are inserted between them and their sum exceeds their number by $1$. Find the number of means inserted.

The ratio of second to seventh of $n$ $A.M.\mathrm{s}$ between $-7$ and $65$ is $1:7$. Find $n$.

Insert six arithmetic means between $2$ and $16$. Also prove that their sum is $6$ times the $A.M.$ between $2$ and $16$.

From the below given distribution find out mean.

$\begin{array}{cccccc}x:& 18& 29& 10& 11& 12\\ f:& 7& 2& 15& 10& 5\end{array}$

The arithmetic mean of the following discrete data $12, 14, 20, 23, 25, 32$ is given by

The arithmetic mean of five natural numbers is $40 .$ The largest exceeds the smallest number by $10 .$ If $\alpha$ is the maximum possible value for the largest of these $5$ numbers, then the number of positive integral divisors of $\alpha$ is ______

The mean marks of $25$ boys in a class is $61$ and the mean marks of $35$ girls in the same class is $58.$ Then the mean of all $60$ students is

The arithmetic mean of first $n$ odd natural number is

If $\mathrm{A}.\mathrm{M}.$ between $p^{\mathrm{th}}$ and $q^{\mathrm{th}}$ terms of an $\mathrm{A}.\mathrm{P}.$ be equal to the $\mathrm{A}.\mathrm{M}.$ between $r^{\mathrm{th}}$ and $s^{\mathrm{th}}$ terms of the $\mathrm{A}.\mathrm{P}.$, then $p+q$ is equal to

A car covers a distance of $65$ km in first hour of its journey, $55$ km in second hour and $90$ km in the last hour. Find the average speed of the car for the whole journey.

A series has $6$ consecutive odd numbers such that the last number of the series is one-third of $429$. Find the arithmetic mean of the series.

Find the arithmetic mean of three numbers such that the first number is $40\%$ more than the second number and also $30\%$ less than the third number. Third number is $50$.