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# help

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Which of the following is largest?

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

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.)

Jun 25, 2020
edited by amazingxin777  Jun 25, 2020
edited by amazingxin777  Jun 26, 2020
#2
-1

a=998*999^1000;printa;
b=999*999^999;printb;
c=997*999^1001;printc;
d=996*998^1000;printd

RESULTS:
C = 3.662 257 462 E+3005
A = 3.669 600 339 E+3002
D = 1.345 242 644 E+3002
B = 3.676 954 248 E+2999

Jun 25, 2020