What is the value of $$\ \sqrt{1+\frac{1}{2^{2}}+\frac{1}{3^{2}}}+\sqrt{1+\frac{1}{3^{2}}+\frac{1}{4^{2}}}+\sqrt{1+\frac{1}{4^{2}}+\frac{1}{5^{2}}}\ $$?

Expression = $$\ \sqrt{1+\frac{1}{2^{2}}+\frac{1}{3^{2}}}+\sqrt{1+\frac{1}{3^{2}}+\frac{1}{4^{2}}}+\sqrt{1+\frac{1}{4^{2}}+\frac{1}{5^{2}}}\ $$

= $$\ \sqrt{1+\frac{1}{4}+\frac{1}{9}}+\sqrt{1+\frac{1}{9}+\frac{1}{16}}+\sqrt{1+\frac{1}{16}+\frac{1}{25}}\ $$

= $$\sqrt{\frac{36+9+4}{36}}+\sqrt{\frac{144+16+9}{144}}+\sqrt{\frac{400+25+16}{400}}$$

= $$\sqrt{\frac{49}{36}}+\sqrt{\frac{169}{144}}+\sqrt{\frac{441}{400}}$$

= $$\frac{7}{6}+\frac{13}{12}+\frac{21}{20}$$

= $$\frac{70+65+63}{60}=\frac{198}{60}=\frac{33}{10}$$

=> Ans - (D)