Question

Which is larger, $(1.01{)}^{1000000}or10000?$

Answer 1

We have

$(1.01{)}^{1000000}$ or 10000

$=(1.01{)}^p-10000$, where 1000000 =p

$=(1+0.01{)}^p-10000$

${=}^p{C}_0{+}^p{C}_1\times 0.01{+}^p{C}_2\times (0.01{)}^2+....{+}^p{C}_p\times (0.01{)}^p-10000$

$=1+(p\times 0.01)+{[}^p{C}_2\times (0.01{)}^2+...+(0.01{)}^2+.....+\left(1.01{)}^{1000000}\right]-10000$

=1+10000+[a positive real number]-10000

=1+(a positive real number)>0

Hence $(1.01{)}^{1000000}>10000$