Question

Using binomial theorem, indicate which is larger (1.1)¹⁰⁰⁰⁰ or 1000.

Answer 1

We have:
(1.1)^{10000
$$\begin{gathered} =(1+0.1{)}^{10000} \\ ={}^{10000}C_0\times {\left(0.1\right)}^0+{}^{10000}C_1\times (0.1{)}^1+{}^{10000}C_2\times (0.1{)}^2+...{}^{10000}C_{10000}\times (0.1{)}^{10000} \\ =1+10000\times 0.1+\mathrm{other}\mathrm{positive}\mathrm{terms} \\ =1+10000+\mathrm{other}\mathrm{positive}\mathrm{terms} \\ =10001+\mathrm{other}\mathrm{positive}\mathrm{terms} \\ \because 10001>1000 \\ \therefore (1.1{)}^{10000}>1000 \end{gathered}$$}