1/(√1+√2)+1/(√2+√3)+1/(√3+√4)+⋯+1/(√899+√900) 13/08/2021 Bởi Delilah 1/(√1+√2)+1/(√2+√3)+1/(√3+√4)+⋯+1/(√899+√900)
Đáp án: $\dfrac{1}{\sqrt{1}+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+….+\dfrac{1}{\sqrt{899}+\sqrt{900}}=29$ Giải thích các bước giải: Ta có: $\begin{split}\dfrac{1}{\sqrt{n-1}+\sqrt{n}}&=\dfrac{\sqrt{n-1}-\sqrt{n}}{(\sqrt{n-1}-\sqrt{n})(\sqrt{n-1}+\sqrt{n})}\\&=\dfrac{\sqrt{n-1}-\sqrt{n}}{n-1-n}\\&=-\sqrt{n-1}+\sqrt{n}\end{split}$ $\dfrac{1}{\sqrt{1}+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+….+\dfrac{1}{\sqrt{899}+\sqrt{900}}$ $=-\sqrt{1}+\sqrt{2}-\sqrt{2}+\sqrt{3}+…-\sqrt{899}+\sqrt{900}$ $=\sqrt{900}-1$ $=30-1$ $=29$ Bình luận
Đáp án:
$\dfrac{1}{\sqrt{1}+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+….+\dfrac{1}{\sqrt{899}+\sqrt{900}}=29$
Giải thích các bước giải:
Ta có:
$\begin{split}\dfrac{1}{\sqrt{n-1}+\sqrt{n}}&=\dfrac{\sqrt{n-1}-\sqrt{n}}{(\sqrt{n-1}-\sqrt{n})(\sqrt{n-1}+\sqrt{n})}\\&=\dfrac{\sqrt{n-1}-\sqrt{n}}{n-1-n}\\&=-\sqrt{n-1}+\sqrt{n}\end{split}$
$\dfrac{1}{\sqrt{1}+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+….+\dfrac{1}{\sqrt{899}+\sqrt{900}}$
$=-\sqrt{1}+\sqrt{2}-\sqrt{2}+\sqrt{3}+…-\sqrt{899}+\sqrt{900}$
$=\sqrt{900}-1$
$=30-1$
$=29$