Sequences and Series

Let α and β be the roots of equation $x^2-6x-2=0$. If $a_n=\alpha^n-\beta^n$, for $n\ge1$, then the value of $\dfrac{a_{10}-2a_8}{2a_9}$ is equal to :

If $m$ is the A.M. of two distinct real numbers $l$ and $n$ ($l,n>1$) and $G_1$, $G_2$ and $G_3$ are three geometric means between $l$ and $n$, then $G_1^4+2G_2^4+G_3^4$ equals.

The sum of first 9 terms of the series
$$\frac{1^3}1+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\dots$$
is :

The least value of the product $xyz$ for which the determinant $\begin{vmatrix}x&1&1\\1&y&1\\1&1&z\end{vmatrix}$ is non-negative, is :

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 :

The value of $\displaystyle\sum_{r=16}^{30}(r+2)(r-3)$ is equal to :

The sum of the 3$^{\text{rd}}$ and the 4$^{\text{th}}$ terms of a G.P. is 60 and the product of its first three terms is 1000. If the first term of this G.P. is positive, then its 7$^{\text{th}}$ term is :

If $\displaystyle\sum_{n=1}^5\frac1{n(n+1)(n+2)(n+3)}=\frac k3$, then k is equal to :

Let $a,b,c$ be positive integers such that $\dfrac ba$ is an integer. If $a,b,c$ are in geometric progression and the arithmetic mean of $a,b,c$ is $b+2$, then the value of
$$\frac{a^2+a-14}{a+1}$$
is

The probability that $x_1$, $x_2$, $x_3$ are in an arithmetic progression, is

Let $\alpha$ and $\beta$ be the roots of equation $px^2+qx+r=0$, $p\ne0$. If $p$, $q$, $r$ are in A.P. and $\dfrac1\alpha+\dfrac1\beta=4$, then the value of $|\alpha-\beta|$ is :

If $(10)^9+2(11)^1(10)^8+3(11)^2(10)^7+\dots+10(11)^9=k(10)^9$, then $k$ is equal to :

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 :

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$^{\text{th}}$ term is :

If the sum
$$\frac{3}{1^2}+\frac{5}{1^2+2^2}+\frac{7}{1^2+2^2+3^2}+\dots\text{ upto 20 terms}$$
is equal to $\dfrac{k}{21}$, then $k$ is equal to :

The coefficient of $x^{50}$ in the binomial expansion of
$$(1+x)^{1000}+x(1+x)^{999}+x^2(1+x)^{998}+\dots+x^{1000}$$
is :

In a geometric progression, if the ratio of the sum of first 5 terms to the sum of their reciprocals is 49, and the sum of the first and the third term is 35. Then the first term of this geometric progression is :

The sum of the first 20 terms common between the series $3+7+11+15+\dots$ and $1+6+11+16+\dots$, is :

Let G be the geometric mean of two positive numbers a and b, and M be the arithmetic mean of $\dfrac1a$ and $\dfrac1b$. If $\dfrac1M:G$ is 4 : 5, then a : b can be :

The least positive integer n such that $1-\dfrac23-\dfrac2{3^2}-\dots-\dfrac2{3^{n-1}}<\dfrac1{100}$, is :