Which is bigger?

Algebra Level 1

Which of the following is larger?

A = 998 × 99 9 1000 B = 999 × 99 9 999 C = 997 × 99 9 1001 D = 996 × 99 8 1000 \large A = 998 \times 999^{1000} \\ \\ \large B = 999 \times 999^{999} \\ \\ \large C = 997 \times 999^{1001} \\ \\ \large D = 996 \times 998^{1000}

C A B D

Syed Hamza Khalid by Syed Hamza Khalid , 17, France

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1 solution

Naren Bhandari
Oct 15, 2017

A = 998 × 99 9 1000 B = 999 × 99 9 999 C = 997 × 99 9 1001 D = 996 × 99 8 1000 \begin{aligned} & A= 998\times 999^{1000} \\ & B= 999\times 999^{999} \\ & C = 997\times 999^{1001} \\ & D = 996\times 998^{1000}\end{aligned}

Now A B = 998 × 99 9 1000 999 × 99 9 999 = 998 × 999 999 ⟹ A > B \frac{A}{B} = \frac{998\times 999^{1000}}{999\times 999^{999}} = \frac{998\times 999}{999} \implies A> B

A C = 998 × 99 9 1000 997 × 99 9 1001 = 998 997 × 99 ⟹ C > A \frac{A}{C} = \frac{998\times 999^{1000}}{997\times 999^{1001}} = \frac{998}{997\times 99} \implies C> A

C D = 997 × 99 9 1001 996 × 99 8 1000 ⟹ C > D \frac{C}{D} = \frac{997\times 999^{1001}}{996\times 998^{1000}} \implies C>D

Thus, among them C C biggest of all.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Very interesting approach!

Syed Hamza Khalid - 3 years, 7 months ago

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