Sequences and S...

Three bells commenced to toll at the same time and tolled at intervals of $$20, 30, 40$$ seconds respectively. If they toll together at $$6$$ am, then which of the following is the time at which they can toll together

The first term of an AP is $$148$$ and the common difference is $$-2$$. If the AM of first $$n$$ terms of the AP is $$125$$, then the value of $$n$$ is

Find the sum of the series 
$$\displaystyle \frac{1}{2\cdot 3}+\frac {1}{4\cdot 5}+\frac {1}{6\cdot 7}+ ...$$

If the natural numbers are divided into groups of {1}, {2, 3}, {4, 5, 6}, {7, 8, 9, 10} ....Then the /sum of 50th group is 

If $$\displaystyle \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + ..... $$ upto $$\displaystyle \infty = \frac{\pi^2}{6},$$ then $$\displaystyle \frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + .... =$$

The value of a for which side of nth square equals the diagonals of $$(n + 1)^{th}$$ square is 

$$\displaystyle \sum_{k = 1}^{\infty} \dfrac {6^{k}}{(3^{2k + 1} + 2^{2k + 1}) - (3^{k}2^{k + 1} + 2^{k}3^{k + 1})}$$ is equal to

Let $$S = \displaystyle \sum_{n = 1}^{99} = \dfrac {5^{100}}{5^{100} + 25^{n}}$$ then find the value of $$[S]$$, where $$[.] = G.I.F.$$

If $$\alpha =1 / 4$$ and $$P_n$$ denotes the perimeter of the nth square then$$\sum_{n=1}^{\infty } P_n$$ equals 

If $$ a_1 \in R - \left \{ 0 \right \}, i = 1, 2, 3, 4$$ and $$x \in R$$ and $$\left ( \sum_{i=1}^{3}a_i^2\right ) x^2-2x \left ( \sum_{i=1}^{3}a_i a_{i+1}\right )+\sum_{i=2}^{4}a_i^2\leq 0$$, then $$a_1, a_2, a_3, a_4$$ are in