Summer Problem Solving Marathon Solution #17

Let x be the number that is added. Then 20 + x, 50 + x, and 100 + x form a geometric sequence. Let r be the common ratio of that sequence. Then:

50 + x = (20 + x)r
100 + x = (50 + x)r

From the first equation, we have r = ^{(50 + x)}⁄_{(20 + x)}. From the second, we have r = ^{(100 + x)}⁄_{(50 + x)}. So we have:

^{(50 + x)}⁄_{(20 + x)} = ^{(100 + x)}⁄_{(50 + x)}

Cross-multiplying:

(50 + x)(50 + x) = (20 + x)(100 + x)

Or:

x² + 100x + 2500 = x² + 120x + 2000

20x = 500

x = 25

So r = (50 + 25)/(20 + 25) = 75/45 = 5/3.