Using binomial theorem, indicate which number is larger: $(1.1)^{10000}$ or $1000$

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Solution

$(1.1)^{10000} = (1 + 0.1)^{10000}$

$= 1 + 10000_{C_{1}} (0.1) + 10000_{C_{2}} (0.1)^{2} + 10000_{C_{3}} (0.1)^{3} + . . .$

$= 1 + 10000 (0.1) + other positive numbers$

$= 1 + 1000 + other positive numbers$

which is greater than 1000.

Thus $(1.1)^{10000} > 1000$ .

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