Cho một cấp số cộng

D

- Hướng dẫn giải

Đáp án D

Ta có

$$\begin{gathered} \left\{\begin{array}{l}{u}_1=1\\ S_{100}=24850\end{array}\right.\iff \left\{\begin{array}{l}{u}_1=1\\ 100\left(2u_1+99d\right)=24850\end{array}\right. \\ \Rightarrow \left\{\begin{array}{l}{u}_1=1\\ d=5\end{array}\right.\Rightarrow \frac{u_n-u_{n-1}}{5}=1 \end{gathered}$$ 

Khi đó 

$$\begin{gathered} 5S=\frac{5}{u_1{u}_2}+\frac{5}{u_2{u}_3}+\mathrm{...}+\frac{5}{u_{49}{u}_{50}} \\ =\frac{u_2-u_1}{u_1{u}_2}+\frac{u_3-u_2}{u_2{u}_3}+\mathrm{...}+\frac{u_{50}-u_{49}}{u_{49}{u}_{50}} \\ =\frac{1}{u_1}-\frac{1}{u_2}+\frac{1}{u_2}-\frac{1}{u_3}+\mathrm{...}+\frac{1}{u_{49}}-\frac{1}{u_{50}}=1-\frac{1}{u_{50}} \\ =1-\frac{1}{u_1+49d}=\frac{245}{246}\Rightarrow S=\frac{49}{246} \end{gathered}$$