Question

cho cấp số cộng Un có U1=1, U2020=10000

tính Sn=\(\dfrac{1}{\sqrt{u1}+\sqrt{u2}}+\dfrac{1}{\sqrt{u2}+\sqrt{u3}}+...+\dfrac{1}{\sqrt{u2019}+\sqrt{u2020}}\)

Answer 1

Công sai \(d=\dfrac{u_{2020}-u_1}{2019}=\dfrac{3333}{673}\).

Ta có \(d.S_n=\dfrac{u_2-u_1}{\sqrt{u_1}+\sqrt{u_2}}+\dfrac{u_3-u_2}{\sqrt{u_2}+\sqrt{u_3}}+...+\dfrac{u_{2020}-u_{2019}}{\sqrt{u_{2019}}+\sqrt{u_{2020}}}=\sqrt{u_2}-\sqrt{u_1}+...+\sqrt{u_{2020}}-\sqrt{u_{2019}}=\sqrt{u_{2020}}-\sqrt{u_1}=100-1=99\)

\(\Rightarrow S_n=\dfrac{99}{d}=\dfrac{2019}{101}\).