Finite Sequences

If $a,\left(b-a\right),\left(c-a\right)$ are in GP and $a=\frac{b}{3}=\frac{c}{5}$ then $a,b,c$ are in

Find the harmonic mean of the following data: $2,4,5,11,14$.

Find the harmonic mean between $\frac{a}{(1-ab)}\text{ and }\frac{a}{(1+ab)}$.

Find the harmonic mean between $7\text{ and }9$

Four numbers are in A.P. If sum of numbers is $50$ and largest number is four times the smaller one, then find the numbers.

If three consecutive terms of $A.P$ are $\frac{4}{5},a,2$, then find the value of $a$

Make the following figures with match sticks and write down the number of match sticks required for each figure. Can you find a common difference in members of the list? Does the list of these numbers form an AP?

If $18$, $a$, $b$, $-3$ are in A.P then $a+b=$

Check the following list of numbers for an A.P. If they form an A.P.,  then find its common difference and write the next four terms.

$a$, $2a$, $3a$, $4a$, .....

Check the following list of numbers for an A.P. If any of them form an A.P., then find its common difference and write the next four terms.

$\sqrt{2}$, $\sqrt{8}$, $\sqrt{18}$, $\sqrt{32}$, .....

Find the first term $\mathrm{a}$ and common difference $\mathrm{d}$ for the following A.P.

$3, -2, -7, -12, ...$

Find the first term $\mathrm{a}$ and common difference $\mathrm{d}$ for the following A.P.

$1, -2, -5, -8, ...$

Find the first term $\mathrm{a}$ and common difference $\mathrm{d}$ for the following A.P.

$\frac{3}{2}, \frac{1}{2}, -\frac{1}{2}, -\frac{3}{2}, ...$

Find the first term $\mathrm{a}$ and common difference $\mathrm{d}$ for the following A.P.

$-7, -9, -11, -13, ...$

Find the first term $\mathrm{a}$ and common difference $\mathrm{d}$ for the following A.P.

$6, 9, 12, 15, ...$

The geometric mean of $\left(x_1, x_2, x_3............x_n\right)$ is $x$ and the geometric mean of $\left(y_1, y_2, y_3.............y_n\right)$ is $y$. Which of the following is/are correct?

$\left(1\right)$ The geometric mean of $\left(x_1{y}_1, x_2{y}_2, x_3{y}_3........x_n{y}_n\right)$ is $xy$

$\left(2\right)$ The geometric mean of $\left(\frac{x_1}{y_1}, \frac{x_2}{y_2}, \frac{x_3}{y_3},....\frac{x_n}{y_n}\right)$ is $\frac{x}{y}$.

Let $\alpha$ and $\beta$ be two numbers where $\alpha <\beta$$.$ The geometric mean of these numbers exceeds the smaller number $\alpha$ by $12$ and the arithmetic mean of the same numbers is smaller by $24$ than the larger number $\beta$$,$ then the value of $|\beta -\alpha |$ is

Evaluate $1\mathrm{ }+\frac{4}{5}\mathrm{ }+\frac{7}{25}\mathrm{ }+\frac{10}{125}+\dots$ to infinite terms.

Find the value up to infinity $\mathrm{ }{2}^{\frac{1}{4}} .4^{\frac{1}{8}} .\mathrm{ }{8}^{\frac{1}{16}}\dots \mathrm{\infty }$

If ${10}^9+2\left(11\right){\left(10\right)}^8+3.{\left(11\right)}^2{\left(10\right)}^7+\dots +10{\left(11\right)}^9=\mathrm{k}{\left(10\right)}^9,\mathrm{ }$ then value of $\mathrm{ }\mathrm{ }\mathrm{k}\mathrm{ }$ is equal to