$a)$   $a=2n^2$  

$a_9=2\cdot 9^2=2\cdot 81=162$  

$a_{11}=2\cdot {11}^2=2\cdot 121=242$  

$a_{k+1}=2\cdot {\left(k+1\right)}^2=2\cdot \left(k^2+2k+1\right)=2k^2+4k+2$  

$a_{3m}=2\cdot {\left(3m\right)}^2=2\cdot 9m^2=18m^2$  

$a_{m+k}=2\cdot {\left(m+k\right)}^2=2\cdot \left(m^2+2mk+k^2\right)=2m^2+4mk+2k^2$  

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$b)$   $a=\frac{1}{n}$  

$a_9=\frac{1}{9}$  

$a_{11}=\frac{1}{11}$  

$a_{k+1}=\frac{1}{k+1}$  

$a_{3m}=\frac{1}{3m}$  

$a_{m+k}=\frac{1}{m+k}$

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$c)$   $a=\frac{{\left(-1\right)}^n}{n}$  

$a_9=\frac{{\left(-1\right)}^9}{9}=\frac{-1}{9}=-\frac{1}{9}$  

$a_{11}=\frac{{\left(-1\right)}^{11}}{11}=\frac{-1}{11}=-\frac{1}{11}$  

$a_{k+1}=\frac{{\left(-1\right)}^{k+1}}{k+1}$  

$a_{3m}=\frac{{\left(-1\right)}^{3m}}{3m}$  

$a_{m+k}=\frac{{\left(-1\right)}^{m+k}}{m+k}$

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$d)$   $a=\frac{4n-3}{n+5}$  

$a_9=\frac{4\cdot 9-3}{9+5}=\frac{36-3}{14}=\frac{33}{14}$  

$a_{11}=\frac{4\cdot 11-3}{11+5}=\frac{44-3}{16}=\frac{41}{16}$  

$a_{k+1}=\frac{4\cdot \left(k+1\right)-3}{k+1+5}=\frac{4k+4-3}{k+6}=\frac{4k+1}{k+6}$  

$a_{3m}=\frac{4\cdot \left(3m\right)-3}{3m+5}=\frac{12m-3}{3m+5}$  

$a_{m+k}=\frac{4\cdot \left(m+k\right)-3}{m+k+5}=\frac{4m+4k-3}{m+k+5}$