If $a, b, c, d$ and $p$ are different real numbers such that $(a^2 + b^2 + c^2)p^2 - 2(ab + bc + cd)p + (b^2 + c^2 + d^2) \le 0$,then $a, b, c, d$ are in

Which term of the sequence $\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \dots$ is $\frac{1}{19683}$?

Write the first five terms of the sequence whose $n^{th}$ term is $a_{n} = 2^{n}$.

The sum of some terms of a geometric progression is $728$. If the common ratio is $3$ and the last term is $486$,what is the first term of the progression?

Let the positive numbers $a_1, a_2, a_3, a_4$ and $a_5$ be in a $G$.$P$. Let their mean and variance be $\frac{31}{10}$ and $\frac{m}{n}$ respectively,where $m$ and $n$ are co-prime. If the mean of their reciprocals is $\frac{31}{40}$ and $a_3+a_4+a_5=14$,then $m+n$ is equal to $.........$.

What will $Rs. 500$ amount to in $10$ years after its deposit in a bank which pays an annual interest rate of $10\%$ compounded annually?