Sequences and Series

There are four numbers of which the first three are in GP and the last three are in AP, whose common difference is 6 . If the first and the last numbers are equal, then two other numbers are

If $\sum_{k=1}^n k(k+1)(k-1)=p n^4+q n^3+t n^2+s n$ where $p, q, t$ and $s$ are constants, then the value of $s$ is equal to

If $a\gt0, b\gt0, c\gt0$ and $a, b, c$ are distinct, then $(a+b)(b+c)(c+a)$ is greater than

If $\tan ^{-1}\left[\frac{1}{1+1.2}\right]+\tan ^{-1}\left[\frac{1}{1+2.3}\right]+$ $\ldots+\tan ^{-1}\left[\frac{1}{1+n(n+1)}\right]=\tan ^{-1}[x]$, then $x$ is equal to

The sum of the infinite series $1+\frac{5}{6}+\frac{12}{6^2}+\frac{22}{6^3}+\frac{35}{6^4}+\frac{51}{6^5}+\frac{70}{6^6}+\ldots .$. is equal to:

If $A=1+r^a+r^{2 a}+r^{3 a}+\ldots \infty$ and $\mathrm{B}=1+\mathrm{r}^{\mathrm{b}}+\mathrm{r}^{2 \mathrm{~b}}+\mathrm{r}^{3 \mathrm{~b}}+\ldots \infty$, then $\frac{\mathrm{a}}{\mathrm{b}}$ is equal to

If $y=\tan ^{-1} \frac{1}{x^2+x+1}+\tan ^{-1} \frac{1}{x^2+3 x+3}$ $+\tan ^{-1} \frac{1}{x^2+5 x+7}+\ldots$.to $n$ terms, then $\frac{d y}{d x}=$

If arithmetic mean of two distinct positive real number $a$ and $b(a \gt b)$ be twice their geometric mean, then $a: b=$

If the sum of an infinite GP $a, a r, a r^2, a r^3, \ldots$ is 15 and the sum of the squares of its each term is 150 , then the sum of $a r^2, a r^4, a r^6, \ldots$ is :

If $\alpha, \beta, \gamma \in[0, \pi]$ and if $\alpha, \beta, \gamma$ are in AP, then $\frac{\sin \alpha-\sin \gamma}{\cos \gamma-\cos \alpha}$ is equal to

In a sequence of $21$ terms, the first $11$ terms are in $AP$ with common difference $2$ and the last $11$ terms are in $GP$ with common ratio $2$. If the middle term of $AP$ be equal to the middle term of the $GP$,…

Let $a_1, a_2, a_3 \ldots$ be a harmonic progression with $a_1=5$ and $a_{20}=25$. The least positive integer $n$ for which $a_n < 0$, is