If [ x ] represents the greatest integer not greater than x , then 1 + 1 100000 100000 =

Let $$\begin{gathered} A=\left[\right(1+\frac{1}{10000}{)}^{10000}] \\ =[(1+0.0001{)}^{1000}] \end{gathered}$$

We know that,

$$\begin{gathered} (1+x{)}^n={}^{n}C_0+{}^{n}C_1·x+{}^{n}C_2·x^2+{}^{n}C_n·x^n \\ A=[{}^{1000}C_0+{}^{1000}C_1·(0.0001)+{}^{1000}C_2·(0.0001{)}^2+......+{}^{1000}C_{1000}·(0.0001{)}^n] \\ A=[1+10000\times 0.0001+\frac{10000\times 999}{2}\times {10}^{-8+........]} \\ A=[1+1+0.49+...] \\ A=[2.049+...] \\ A=2. \end{gathered}$$