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**

**S**equence

Series

Real Analysis

## Check the Answer

But try the problem first…

Answer: is 840.

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.

## Other useful links

- https://www.cheenta.com/cubes-and-rectangles-math-olympiad-hanoi-2018/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s

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