Which of the following is largest?

\(\large A = 998 \times 999^{1000} \\ \\ \large B = 999 \times 999^{999} \\ \\ \large C = 997 \times 999^{1001} \\ \\ \large D = 996 \times 998^{1000}\)

Guest Jun 25, 2020

#1**+5 **

first compare A and B. At first glance we con obviously tell that A is larger because B is only \(999^{1000}\)while A is 998 \(\times\) B. So we have one inequality already A > B.

Now let's compare B and C. B is \(999^{1000}\) and C is definetely greater than that so another inequality:

C > B.

Now let's compare C and D: If you estimate, then C is greater.

C > D.

but we aren't done!!! A may be greater than B but it may be smaller than C or D. When we compare A and C then C is greater by a lot.

so C > A.

Now we compare all our inequalities: B < A < C and D < C.

Let's compare A and D. and we found that A is larger.

D < A.

Last one: is D smaller than B? D is in fact smaller than B.

This is the final solution: \(\boxed{D < B < A < C}\)

Yikes!! (I may have done some estimation wrong.)

amazingxin777 Jun 25, 2020