Sequences and Series

$\frac{1}{2 \cdot 5}+\frac{1}{5 \cdot 8}+\frac{1}{8 \cdot 11}+\ldots \frac{1}{(3 n-1)(3 n+2)}=$

The angles $A, B, C$ of a triangle $A B C$ are in AP. If $A B=6, B C=7$, then $A C=$

$p, x_1, x_2 \ldots, x_n$ and $q, y_1, y_2, \ldots, y_{\mathrm{n}}$ are two arithmetic progressions with common differences a and $\mathrm{b}$ respectively. If $\alpha$ and $\beta$ are the arithmetic means of $x_1, x_2,…

The sum of the cubes of three consecutive natural numbers is divisible by

If the angles $A, B$ and $C$ of $\triangle A B C$ are in arithmetic progression, then

If the altitude of a triangle are in arithmetic progression, then the sides of the triangles are in

If $a, b, c$ are in AP, $b-a, c-b$ and $a$ are in GP, then $a: b: c$ is

If the roots of the equation $4 x^3-12 x^2+11 x+m=0$ are ir arithmetic progression, then $m=$

The number of pairs of consecutive positive even integers such that the sum of their squares is 290 is

If the roots of the equation $x^3+a x^2+b x+c=0$ are in arithmetic progression, then

For $n \in N$, what is the value of $1+4+7+\cdots+(3 n-2)$

For all $n \in N, \frac{3^n-1}{2} \geq$