Question

Using binomial theorem determine which number is larger $(1.2{)}^{4000}$ or 800?

Answer 1

$(1.2{)}^{4000}=(1+0.2{)}^{4000}$

${=}^{4000}{C}_0\left(0.2{)}^0\right(1{)}^{4000}{C}_1\times (0.2{)}^1\times 1^{3999}+....{+}^{4000}{C}_{400}(0.2{)}^{4000}{1}^0$

$=1+4000\times 0.2\times 1+....+\left({0.2}^{4000}\right)$

$=1+800+...+(0.2{)}^{4000}$

$\therefore 801>800$

Here, we dearly observe $(1,2{)}^{4000}$ is less than (801) thus, $(1.2{)}^{4000}<800.$