Question

Prove that $\sqrt{1}0\left\{\right(\sqrt{1}0+1{)}^{100}-(\sqrt{1}0-1{)}^{100}\}$ is a whole number.

Answer 1

We have

$\left\{\right(\sqrt{1}0+1{)}^{100}-(\sqrt{1}0-1{)}^{100}\}$

$=2\{{}^{100}{C}_1.(\sqrt{10}){+}^{100}{C}_3.(\sqrt{10}{)}^3+......{+}^{100}{C}_{99}.\left(\sqrt{10}{)}^{99}\right\}$

$=2\sqrt{10}\{{}^{100}{C}_1.{+}^{100}{C}_3.(\sqrt{10}{)}^2+......{+}^{100}{C}_{99}.\left(\sqrt{10}{)}^{98}\right\}$.......(1).

$\therefore \sqrt{10}\left\{\right(\sqrt{1}0+1{)}^{100}-(\sqrt{1}0-1{)}^{100}\}$

$=20\{{}^{100}{C}_1.{+}^{100}{C}_3.(\sqrt{10}{)}^2+......{+}^{100}{C}_{99}.\left(\sqrt{10}{)}^{98}\right\}$ [Using (1)].

Each term inside and outside of the brackets are whole and hence their addition and multiplication is going to be a whole number.