Arithmetic Mean

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

Prove that the ratio of the sums of $\mathrm{p} \mathrm{A}.\mathrm{M}$’s and $\mathrm{q} \mathrm{A}.\mathrm{M}.$'s between two numbers is $\mathrm{p} : \mathrm{q}$.

What is the sum of  $\mathrm{n} \mathrm{A}.\mathrm{M}.$’s between a and b?

Determine the A.M. between $5\frac{1}{4}$ and $7\frac{3}{4}$.

It is given that ${\log }_{a}n, {\log }_{b}n, {\log }_{c}n$ are in $A.P.$, prove that $c^2=(ac{)}^{\log ab}$

Solve for $x:(x-1{)}^3+(x-2{)}^3+(x-3{)}^3+(x-4{)}^3+(x-5{)}^3=0$

If $a, b, c$ are in $\mathrm{A}.\mathrm{P}., p, q, r$ are in $\mathrm{H}.\mathrm{P}.$ and $ap, bq, cr$ are in $\mathrm{G}.\mathrm{P}.;$ show that $\frac{p}{r}+\frac{r}{p}=\frac{a}{c}+\frac{c}{a}$

Suppose $x, y, z$ are in $\mathrm{A}.\mathrm{P}.$ and that $\sin (y+z)=x, \sin (z+x)=y$ and $\sin (x+y)=z;$ prove that $\tan x, \tan y, \tan z$ are in $\mathrm{A}.\mathrm{P}.$ [Assume, $\sin 2y\ne 0$]

If $x, y, z$ are positive and $x+y+z=1,$ prove that $\left(\frac{1}{x}-1\right)\left(\frac{1}{y}-1\right)\left(\frac{1}{z}-1\right)\ge 8$.

[Hints: Use the result that, for $+\mathrm{ve}$ integers  $\mathrm{A}.\mathrm{M}.\ge \mathrm{G}.\mathrm{M}.$]

If $\frac{1}{a}+\frac{1}{c}+\frac{1}{a-b}+\frac{1}{c-b}=0$ prove that $a, b, c$ in $\mathrm{H}.\mathrm{P}.$ or $a, \frac{b}{2}, c$ are in $\mathrm{A}.\mathrm{P}.$

If $\log \left(\frac{5c}{a}\right), \log \left(\frac{3b}{5c}\right)$ and $\log \left(\frac{a}{3b}\right)$ are in $\mathrm{A}.\mathrm{P}$. where $a, b, c$, are in $\mathrm{G}.\mathrm{P}$., then show that $a, b$ and $c$ cannot be lengths of three sides of a triangle.

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 $50$.

If one geometric mean $G$, and two arithmetic means $p, q$, be inserted between two numbers, then prove that, $G^2=\left(2p-q\right)\left(2q-p\right)$.

Which of the following is not an absolute measure of dispersion?

In a race, the first four winners are to be awarded points .Each winner’s points must be 5 more than that of the next position winner. Total sum of the points to be awarded is 50. What will be the points for the third position winner?

Four quantities are such that their arithmetic mean (A.M) is the same as the A.M of the first three quantities. The fourth quantity is:

For how many days, the rainfall was less than the mean rainfall?

Find the mean rainfall for the week: