Determined the value of 1 1 + 2 + 1 2 + 3 \, + 1 3 + 4 \, + ...... + 1 120 + 121 = ?

Answer: Option B

Given expressing,
$$=\frac{1}{\sqrt{1}+\sqrt{2}}\text{ + }\frac{1}{\sqrt{2}+\sqrt{3}}$$     $$+\frac{1}{\sqrt{3}+\sqrt{4}}$$   $$+......+$$   $$\frac{1}{\sqrt{120}+\sqrt{121}}$$
$$=\frac{1}{\sqrt{2}+\sqrt{1}}\text{ + }\frac{1}{\sqrt{3}+\sqrt{2}}$$     $$+\frac{1}{\sqrt{4}+\sqrt{3}}$$   $$+......+$$   $$\frac{1}{\sqrt{121}+\sqrt{120}}$$
$$=\frac{1}{\sqrt{2}+\sqrt{1}}\times$$   $$\frac{\sqrt{2}-\sqrt{1}}{\sqrt{2}-\sqrt{1}}\text{ + }$$   $$\frac{1}{\sqrt{3}+\sqrt{2}}\times$$   $$\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}+$$   $$\frac{1}{\sqrt{4}+\sqrt{3}}\times$$   $$\frac{\sqrt{4}-\sqrt{3}}{\sqrt{4}-\sqrt{3}}+$$   $$......+$$   $$\frac{1}{\sqrt{121}+\sqrt{120}}\times$$   $$\frac{\sqrt{121}-\sqrt{120}}{\sqrt{121}-\sqrt{120}}$$
$$=\frac{\sqrt{2}-\sqrt{1}}{2-1}+\frac{\sqrt{3}-\sqrt{2}}{3-2}$$     $$+\frac{\sqrt{4}-\sqrt{3}}{4-3}$$   $$+......+$$   $$\frac{\sqrt{121}-\sqrt{120}}{121-120}$$
$$=\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}$$     $$+\sqrt{4}-\sqrt{3}$$   $$+......+$$   $$\sqrt{121}-\sqrt{120}$$
$$=-1+\sqrt{121}$$
$$=-1+11$$
$$=10$$