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Question
Using binomial theorem, indicate which of the following two number is larger: `(1.01)^(1000000)`, 10
Solution
`(1.01)^(1000000) = (1 + 0.01)^(1000000)`
= `""^(1000000)"C"_0(1)^(1000000) + ""^(1000000)"C"_1(1)^(999999)(0.01)^1 + ""^(1000000)"C"_2 (1)^(999998)(0.01)^2 + ""^(1000000)"C"_3(1)^(999997)(0.01)^3 + ..........`
= `1(1) + 1000000xx 1/10^2 + (1000000 xx 999999)/2 xx 1/10000 + .........`
= 1 + 10000 + 50 × 999999 + ........ which is > 10000
So `(1.01)^(1000000) > 10000`
(i.e.) `(1.01)^(1000000)` is larger
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