Differentiation to solve modified sum of binomial coefficients

Q.

Find the value of the sum $\sum _{n=1}^{13}(i^n+i^{n+1})$, where $i=\sqrt{-1}$.

1. $i$

2. $i-1$

3. $-i$

4. $0$

Q.

With usual notations, $C_1+2^2{C}_2+3^2{C}_3+.......=$

1. n(n-1)2^n-2

2. (n+1)(n+2)2^n-2

3. n(n+1)2^n-2

4. n(n - 1)2^n-2

Q.

The sum of $(n-1)$ terms of $1+(1+3)+(1+3+5)+...$is

1. $\frac{n(n+1)(2n+1)}{6}$

2. $n^2$

3. $\frac{n(n-1)(2n-1)}{6}$

4. none of these

Q. Statement-1: ${\sum }_{r=0}^n(r+1){}^n{C}_r=(n+2)2^{n-1}.$
Statement-2: ${\sum }_{r=0}^n(r+1){}^n{C}_r{x}^r=(1+x{)}^n+nx(1+x{)}^{n-1}.$

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

Q.

$\underset{x\to -1}{lim}\frac{(1+x)(1-x^2)(1+x^3)(1-x^4)....(1-x^{4n})}{{\left[\right(1+x\left)\right(1-x^2\left)\right(1+x^3\left)\right(1+x^4).......(1-x^{2n}\left)\right]}^2}\text{is equal to:}$

1. 

2. 

3. 

4. 

Q.

If $(1+x{)}^{15}$ = $C_0+C_{1}x+C_2{x}^2+.........+C_{15}{x}^{15}$, then

$C_2+2C_3+3C_4+........+14C_{15}$=

1. 

2. 

3. 

4. None of these

Q.

$\underset{x\to \infty }{lim}{\left(\frac{3x^2+2x+1}{x^2+x+2}\right)}^{\frac{6x+1}{3x+2}}\text{is equal to}$

1. 1

2. 4

3. 3

4. 9

Q.

$P\left(n\right):5^{2n+1}+3^{n+2}{.2}^{n-1}$ is divisible by

1. 19

2. 18

3. 17

4. 14

Q.

$\text{Let f}\left(x\right)=3x^{10}-7x^8+5x^8-21x^3+3x^2-7\text{THEN}{lim}_{h\to 0}\frac{f(1-h)-f\left(1\right)}{h^3+3h}$

1. 53/3

2. 3

3. 50/3

4. 22/3

Q. If $(1+x{)}^{15}=C_0+C_{1}x+C_2{x}^2+......+C_{15}{x}^{15},$ then $C_2+2C_3+3C_4+......+14C_{15}=$

1. None of these
2. 
3. 
4.