Vn method

Q. Let $S_n={\sum }_{k=1}^n\frac{n}{n^2+kn+k^2}and{T}_n={\sum }_{k=0}^{n-1}\frac{n}{n^2+kn+k^2}$ for a = 1, 2, 3, ...... Then,

1. $S_n<\frac{\pi }{3\sqrt{3}}$
2. $S_n>\frac{\pi }{3\sqrt{3}}$
3. $T_n<\frac{\pi }{3\sqrt{3}}$
4. $T_n>\frac{\pi }{3\sqrt{3}}$

Q. In the sequence 1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, ..., where n consecutive terms have the value n, the 1025th term is

1. $2^{11}$
2. $2^9$
3. $2^{10}$
4. $2^8$

Q. The sum 1(1!) + 2(2!) + 3(3!) + ....+n (n!) equals

[AMU 1999; DCE 2005]

1. 3(n!) + n - 3
2. 2(n!) - 2n - 1
3. (n + 1)! - (n - 1)!
4. (n + 1)! - 1

Q. $\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+....\frac{1}{n(n+1)}$ equals
[AMU 1995; RPET 1996; UPSEAT 1999, 2001]

1. $\frac{1}{n(n+1)}$
2. $\frac{n}{n+1}$
3. $\frac{2}{n(n+1)}$
4. $\frac{2n}{n+1}$

Q. The sum of first 10 terms of the series $1\cdot 2\cdot 3+2\cdot 3\cdot 4+3\cdot 4\cdot 5+.........$ is

1. 6006
2. 2970
3. 3990
4. 4290

Q. The sum of (n + 1) terms of $\frac{1}{1}+\frac{1}{1+2}+\frac{1}{1+2+3}+......$ is
[RPET 1999]

1. $\frac{n}{n+1}$
2. $\frac{2n}{n+1}$
3. $\frac{2}{n(n+1)}$
4. $\frac{2(n+1)}{n+2}$