Question 68
a) What is the value of $$
\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{4}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{6}}+\cdots+\frac{1}{\sqrt{99}+\sqrt{100}} ?
$$
b) if $x=\sqrt{1+2 \cdot \sqrt{1+3 \cdot \sqrt{1+4 \cdot \sqrt{1+5 \cdot \sqrt{\ldots}}}}}$ is a real number, what is its value?
\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{4}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{6}}+\cdots+\frac{1}{\sqrt{99}+\sqrt{100}} ?
$$
b) if $x=\sqrt{1+2 \cdot \sqrt{1+3 \cdot \sqrt{1+4 \cdot \sqrt{1+5 \cdot \sqrt{\ldots}}}}}$ is a real number, what is its value?