Vn method

Q.

Prove 1.2 + 2.3 + 3.4 + $\cdots +n(n+1)=\left[\frac{n(n+1)(n+2)}{3}\right].$

Q.

$\frac{1}{2.5}+\frac{1}{5.8}+\frac{1}{8.11}+\cdots +\frac{1}{(3n-1)(3n+2)}=\frac{n}{6n+4}$

Q.

Find the 20th term of the series $2\times 4+4\times 6+6\times 8+\dots \dots +n$ terms.

Q.

Find the sum of $1^2-2^2+3^2-4^2+5^2-6^2.......$?

Q.

Find the sum to n terms

$3\times 8+6\times 11+9\times 14+\dots$

Q.

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

1. 

2. 

3. 

4. 

Q.

Find the sum of the following series up to n terms:

(i) 5 + 55 + 555 + .....

(ii) .6 + .66 + .666 + .....

Q.

Prove $\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+\cdots +\frac{1}{(2n+1)(2n+3)}=\frac{n}{3(2n+3)}$

Q.

$P\left(n\right):1.2+2.3+3.4+...+n(n+1)=\frac{(n+1)(n+2)}{3}$

The statement P(n) is

1. is true for all natural numbers

2. is not true for all natural numbers

3. is true for n=1

4. is true for n=2

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

Q.

$\mathrm{1.2.3}+\mathrm{2.3.4}+\cdots +n(n+1)(n+2)=\frac{n(n+1)(n+2)(n+3)}{4}$

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

Q. Statement - 1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16)+...+(361 + 380 + 400) is 8000.
Statement - 2: ${\sum }_{k=1}^n(k^3-(k-1{)}^3)=n^3,$ for any natural number n.

1. Statement - 1 is true, Statement - 2 is false.
2. Statement - 1 is false, Statement - 2 is true.
3. Statement - 1 is true, Statement - 2 is true, Statement - 2 is a correct explanation for Statement - 1.
4. Statement - 1 is true, Statement - 2 is true, Statement - 2 is not a correct explanation for Statement - 1.

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.

Each term of the sum
$C_0^2+3.C_1^2+5.C_2^2.....(2n+1)c_n^2$ can be written as \((2r+1){}^nC_r^2

1. True

2. False

Q. $1\times 2+2\times 3+3\times 4+4\times 5+...$

Q.

Find the sum upto first 11 terms of the series 1.4.7 + 4.7.10 + 7.10.13 + . . . . . is __

Q.

Find the sum to n terms

$1\times 2\times 3+2\times 3\times 4+3\times 4\times 5+\dots \dots$

Q.

Find the sum upto first 11 terms of the series 1.4.7 + 4.7.10 + 7.10.13 + . . . . . is __

Q.

Find the sum of the first n terms of the series:

3 + 7 + 13 + 21 + 31 + ...

Q.

Find the sum of first 12 terms of the series 2.5 + 5.8 + 8.11+...............is __

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

[AMU 1999; DCE 2005]

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

Q. $1\times 2\times 3+2\times 3\times 4+3\times 4\times 5+...$