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# Geometric Sequence Problem | AIME I, 2009 | Question 1

Try this beautiful problem from American Invitational Mathematics Examination I, AIME I, 2009 based on geometric sequence. Use hints to solve the problem.

Try this beautiful problem from American Invitational Mathematics Examination I, AIME I, 2009 based on geometric sequence.

## Geometric Sequence Problem – AIME 2009

Call a 3-digit number geometric if it has 3 distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers.

• is 500
• is 250
• is 840
• cannot be determined from the given information

### Key Concepts

Sequence

Series

Real Analysis

But try the problem first…

Source

AIME, 2009

Introduction to Real Analysis, 4th Edition  by Robert G. Bartle, Donald R. Sherbert

## Try with Hints

First hint

3-digit sequence a, ar, $$ar^{2}$$. The largest geometric number must have a<=9.

Second Hint

ar $$ar^{2}$$ less than 9 r fraction less than 1 For a=9 is $$\frac{2}{3}$$ then number 964.

Final Step

a>=1 ar and $$ar^{2}$$ greater than 1 r is 2 and number is 124. Then difference 964-124=840.

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