Maths

$t_n=\frac{3^n-2^n}{3^n}=1-{\left(\frac{2}{3}\right)}^n$

$S_{100}=100-\frac{\frac{2}{3}\left({\left(\frac{2}{3}\right)}^{100}-1\right)}{\frac{2}{3}-1}=100+2.\frac{\left(2^{100}-3^{100}\right)}{3^{100}}$

$=100-2+\frac{2^{101}}{3^{100}}=98+{\left(\frac{2}{3}\right)}^{100}.2<98+1$

$\left[\begin{array}{ccc}\hfill a_1\hfill & \hfill a_2\hfill & \hfill a_3\hfill \\ \hfill b_1\hfill & \hfill b_2\hfill & \hfill b_3\hfill \\ \hfill c_1\hfill & \hfill c_2\hfill & \hfill c_3\hfill \end{array}\right]a_1,b_1,c_{1}t\left\{-1,0,1\right\}$

$\underset{9times}{\underset{?}{a_1+a_2+a_3+....}}=5$

Total = 414

$\lambda =1+\mu$

$\begin{array}{l}\underset{\_}{2\lambda =3\lambda \mu }\\ \lambda =2\to S\left(2,4,6\right)\end{array}$

$\mu =1$

$\overrightarrow{n}=\left|\begin{array}{ccc}\hfill \widehat{i}\hfill & \hfill \widehat{j}\hfill & \hfill \widehat{k}\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 5\hfill \end{array}\right|=\left(7,-2,-1\right)$

7x – 2y – z + d = 0

(2, 4, 6) Þ d = -14 + 8 + 6

d = 0

7x – 2y – z = 0

7 – 2 = 5

g (1) =?

$=f\left(f\left(f\left(1\right)\right)\right)+f\left(f\left(1\right)\right)$

$f\left(1\right)={\left(2\left(1-\frac{1}{2}\right)\left(2+1\right)\right)}^{\frac{1}{50}}=3^{\frac{1}{50}}$

$3^{\frac{1}{50}}+13^{\frac{1}{50}}>1^{\frac{1}{50}}$

3 > 1

$3^{\frac{1}{50}}>2$   $2>3^{\frac{1}{50}}>1$

$3<2^{50}$   $\left[3^{\frac{1}{50}}+1\right]=1+1=2$

${\alpha }_n=19^n-12^n$

$\frac{31{\alpha }_9-{\alpha }_{10}}{57.{\alpha }_2}=\frac{31\left(19^9-12^9\right)-\left(19^{10}-12^{10}\right)}{57\left(19^8-12^8\right)}$

$=\frac{19^9\left(31-19\right)-12^9\left(31-12\right)}{57\left(19^8-12^8\right)}$

$=\frac{19^9.12-12^9.19}{57\left(19^8-12^8\right)}=4$

$P\left(x_1,y_1\right)$

$Q\left(x_2,y_2\right)$

$x_1+x_2=\frac{r}{2},x_1{x}_2=\frac{P}{2}$

$y_1+y_2=s,y_1,y_2=-9$

$\left(x-x_1\right)\left(x-x_2\right)+\left(y-y_1\right)\left(y-y_2\right)=0$

$2\left(x^2+y^2\right)-rx-2sy+p-2q=0$

r = 11, s = 7, p – 2q = -22

${\left|2\left(\overrightarrow{a}*\overrightarrow{b}\right)\right|}^2+4{\left(\overrightarrow{a}.\overrightarrow{b}\right)}^2$

= $4{\left|\overrightarrow{a}\right|}^2{\left|\overrightarrow{b}\right|}^2$

=4 * 16 * 9 = 576

$x_1+x_2=6,13\to 6+6=12$

$\begin{array}{l}{x}_1+x_2=4,11,18\to 4+8+1=13\\ x_1+x_2=2,9,16\to 2+9+3=14\\ x_1+x_2=7,14\to 7+5=12\\ x_1+x_2=5,12\to 5+7=12\end{array}$

= 63

${\displaystyle \sum _{k=1}^{10}\left(2k-1\right).k{.}^{10}{C}_k}$

$=2\sum {k^2}^{10}{C}_k-\sum k{.}^{10}{C}_k$

$x=1\Rightarrow {\displaystyle \sum _{k=0}^{10}k}{.}^{10}{C}_k=10.2^9$

$x=1\Rightarrow \sum k^2{}^{10}{C}_k=90.2^8+10.2^9=2^8\left(90+201\right)=2^8.110$

S = 2⁹ . 110 – 10.2⁹ = 2⁹ . 100

S = 2⁹ . 100

$\frac{2^{11}.25}{2^1-1}$