Vn Method

Q. The value of $\frac{1}{\sqrt{5}+\sqrt{3}}+\frac{1}{\sqrt{7}+\sqrt{5}}+\frac{1}{\sqrt{9}+\sqrt{7}}+\cdots \cdots$ upto $100$ terms is

1. $\frac{200}{\sqrt{218}+\sqrt{12}}$
2. $\frac{100}{\sqrt{203}+\sqrt{3}}$
3. $\frac{100}{\sqrt{203}-\sqrt{3}}$
4. $\frac{200}{\sqrt{218}-\sqrt{12}}$

Q. If $\sum _{k=1}^{360}\left(\frac{1}{k\sqrt{k+1}+(k+1)\sqrt{k}}\right)=a,$ then the value of $19a$ is

Q. Trigonometric series of the form
$\frac{\sin (A-B)}{\cos A\cdot \cos B}+\frac{\sin (B-C)}{\cos B\cdot \cos C}+\frac{\sin (C-D)}{\cos C\cdot \cos D}$
$=\tan A-\tan D$
As we know that,
$\frac{\sin (A-B)}{\cos A\cdot \cos B}=\tan A-\tan B$
Based on the above given information, find $n^{\text{th}}$ term of the series
$\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x}+\cdots$ upto $n$ terms

1. $\frac{1}{2}(\tan 3^{n}x-\tan 3^{n-1}x)$
2. $\frac{1}{2}(\tan 3^{n-1}x-\tan 3^{n}x)$
3. $\frac{1}{2}(\cot 3^{n}x-\cot 3^{n-1}x)$
4. $\frac{1}{2}(\cot 3^{n-1}x-\cot 3^{n}x)$

Q. If $\sum _{r=1}^n{T}_r=\frac{n}{8}(n+1)(n+2)(n+3)$ and $\sum _{r=1}^n\frac{1}{T_r}=\frac{n^2+3n}{4\sum _{k=1}^{p}k}$, then $p$ is equal to

1. $n$
2. $n-1$
3. $n+1$
4. $2n$

Q. If $\sum _{k=1}^{360}\left(\frac{1}{k\sqrt{k+1}+(k+1)\sqrt{k}}\right)=a,$ then the value of $19a$ is

Q. The sum of the series $\frac{1}{3}+\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+\cdots$ upto $n$ terms is $S_n.$ Then $4S_{\infty }$ equals

Q. The sum of the series $\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\dots \dots$ upto $16$ terms is

1. $246$
2. $646$
3. $446$
4. $746$

Q. The sum of $\frac{3}{1\cdot 2}\left(\frac{1}{2}\right)+\frac{4}{2\cdot 3}{\left(\frac{1}{2}\right)}^2+\frac{5}{3\cdot 4}{\left(\frac{1}{2}\right)}^3+\cdots$ upto $20$ terms is

1. $1-\frac{1}{21}{\left(\frac{1}{2}\right)}^{20}$
2. $1+\frac{1}{21}{\left(\frac{1}{2}\right)}^{20}$
3. $1-\frac{1}{20}{\left(\frac{1}{2}\right)}^{21}$
4. $1+\frac{1}{20}{\left(\frac{1}{2}\right)}^{21}$

Q.

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

Q. If $S=\sum _{n=2}^{\infty }\frac{3n^2+1}{(n^2-1{)}^3},$ then $16S$ is

Q. Given $S_1=\sum _{n=0}^k\frac{2n+3}{(n+1{)}^2(n+2{)}^2}$ and $S_2=\sum _{n=1}^k\frac{2}{4n^2-1}$

If $S_1-S_2=\frac{363}{37\times 400},$ then $k=$

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. Let $S=\sum _{n=1}^{9999}\frac{1}{(\sqrt{n}+\sqrt{n+1})(\sqrt[4]{4n}+\sqrt[4]{4n+1})}$, then $S$ is equal to

Q. Find the sum of first $10$ terms of following series
$S=\frac{3\left(1\right)}{1}+\frac{5(1^3+2^3)}{1^2+2^2}+\frac{7(1^3+2^3+3^3)}{1^2+2^2+3^2}+....$

1. $440$
2. $450$
3. $660$
4. $220$

Q. If $S_r$ denotes the sum of the infinite geometric series whose first term is $r$ and common ratio is $\frac{1}{1+r}$, where $r\in N$, then the value of $\sum _{r=1}^{10}{S}_r^2$ is

1. $505$
2. $385$
3. $384$
4. $659$

Q. If $\sum _{k=1}^{n}f\left(k\right)=\frac{n}{2(n+2)}$, then the value of $\sum _{k=1}^{10}\frac{1}{f\left(k\right)}$ is equal to

1. $570$
2. $480$
3. $560$
4. $550$

Q. $\text{List I}$ has four entries and $\text{List II}$ has five entries. Each entry of $\text{List I}$ is to be correctly matched with one or more than one entries of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{Possible value(s) of}\sqrt{i}+\sqrt{-i}\text{is (are)}& \text{(P)}& \sqrt{2}\\ \text{(B)}& \text{If}{z}^3=\overline{z}(z\ne 0),& \text{(Q)}& i\\ & \text{then possible values of}z\text{is/are}& & \\ \text{(C)}& 1+\frac{1}{4}+\frac{1\cdot 3}{4\cdot 8}+\frac{1\cdot 3\cdot 5}{4\cdot 8\cdot 12}+\cdots \cdots \infty & \text{(R)}& \sqrt{2}i\\ \text{(D)}& \frac{1}{3^2+1}+\frac{1}{4^2+2}+\frac{1}{5^2+3}+\cdots \cdots \infty & \text{(S)}& \frac{1}{2}\\ & & \text{(T)}& \frac{13}{36}\end{array}$

Which of the following is CORRECT combination?

1. $\left(C\right)\to \left(P\right);\left(D\right)\to \left(T\right)$
2. $\left(C\right)\to \left(S\right);\left(D\right)\to \left(P\right)$
3. $\left(C\right)\to \left(S\right);\left(D\right)\to \left(T\right)$
4. $\left(C\right)\to \left(P\right);\left(D\right)\to \left(S\right)$

Q. Let $A_1=\left[a_1\right]A_2=\left[\begin{array}{cc}{a}_2& a_3\\ a_4& a_5\end{array}\right]A_3=\left[\begin{array}{ccc}{a}_6& a_7& a_8\\ a_9& a_{10}& a_{11}\\ a_{12}& a_{13}& a_{14}\end{array}\right]\cdots \cdots A_n=[\cdots ]\text{where}{a}_r=\left[{\log }_{2}r\right]$
$\left(\right[\cdot ]$ denotes the greatest integer$)$. Then trace of $A_{10}$ is

Q. Trigonometric series of the form
$\frac{\sin (A-B)}{\cos A\cdot \cos B}+\frac{\sin (B-C)}{\cos B\cdot \cos C}+\frac{\sin (C-D)}{\cos C\cdot \cos D}$
$=\tan A-\tan D$
As we know that,
$\frac{\sin (A-B)}{\cos A\cdot \cos B}=\tan A-\tan B$
Based on the above given information, find sum of the series
$\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x}+\cdots$ upto $n$ terms

1. $\frac{1}{2}(\cot 3^{n}x-\cot x)$
2. $\frac{1}{2}(\tan 3^{n}x-\tan x)$
3. $\frac{1}{2}(\tan 3x-\tan 3^{n-1}x)$
4. $\frac{1}{2}(\cot 3x-\cot 3^{n-1}x)$

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

Q. The $n^{\text{th}}$ term of a sequence of numbers is $a_n$ and given by the formula $a_n=a_{n-1}+2n$ for $n\ge 2$ and $a_1=1.$
The sum of first $20$ terms is

1. $3060$
2. $3059$
3. $2680$
4. $2679$

Q. The sum of the following series
$1+6+\frac{9(1^2+2^2+3^2)}{7}+\frac{12(1^2+2^2+3^2+4^2)}{9}+\frac{15(1^2+2^2+...+5^2)}{11}+...\text{up to}15\text{terms}$, is :

1. $7510$
2. $7820$
3. $7830$
4. $7520$