Which is larger?

Algebra Level 4

Which is larger

$1009 ^ { 2017} \quad \text{ or } \quad 1010 ^ { 1009 } \times 1008 ^ {1008} ?$

Hint: $a_n = \left( 1 + \dfrac{1}{n} \right)^n$ for $n = 1, 2, 3, \ldots$ is an increasing sequence.

1009 ^ { 2017 }

1010 ^ { 1009 } \times 1008 ^ {1008}

1 solution

Chan Lye Lee
Apr 30, 2018

Let $A=1009^{2017}=1009^{1009}\times 1009^{1008}$ and $B=1010 ^ { 1009 } \times 1008 ^ {1008}$ .

Then $\dfrac{A}{B}=\left(\dfrac{1009}{1010}\right)^{1009} \times \left(\dfrac{1009}{1008}\right)^{1008} = \dfrac{\left(1+\dfrac{1}{1008}\right)^{1008}}{\left(1+\dfrac{1}{1009}\right)^{1009}}<1$ as $\left( 1 + \dfrac{1}{n} \right)^n$ is increasing. This means that $A<B$ and hence the result.

Which is larger?

1 solution

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