a) Wyznaczamy pięćdziesiąte wyrazy każdego z danych ciągów.

$a_{50}=\frac{50}{50+1}=\frac{50}{51}$  

$b_{50}={\left(-\sqrt{2}\right)}^{50-1}={\left(-\sqrt{2}\right)}^{49}=-\left({\sqrt{2}}^{49}\right)=-\left(\sqrt{2^{48}\cdot 2}\right)=-2^{24}\sqrt{2}$  

$c_{50}=50\left(50+1\right)=50\cdot 51=2550$  

$d_{50}=5$  

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b) Wyznaczamy dane liczby.

$a_{n+3}=\frac{n+3}{n+3+1}=\frac{n+3}{n+4}$  

$b_{n-1}={\left(-\sqrt{2}\right)}^{n-1-1}={\left(-\sqrt{2}\right)}^{n-2}$  

$c_{n+1}-c_n=\left(n+1\right)\left(n+1+1\right)-n\left(n+1\right)=\left(n+1\right)\left(n+2\right)-n\left(n+1\right)=\left(n+1\right)\left[\left(n+2\right)-n\right]=\left(n+1\right)\cdot 2=2\left(n+1\right)$  

$\frac{d_n+d_{n+1}}{2}=\frac{5+5}{2}=\frac{10}{2}=5$