By adding the same constant to each of the numbers 20, 50, 100, we get a geometric sequence. Find the common ratio of the geometric sequence.
+23246 We need to add the same amount to 20, 50, and 100 to get a geometric sequence.
Call this amount 'x".
Then our terms will be t1 = 20 +x, t2 = 50 + x, and t3 = 100 + x
To be a geometric sequence, the terms t1, t2, and t3 must fit in this proportion:
t3 / t2 = t2 / t1
Substitutng: (100 + x) / (50 + x) = (50 + x) / (20 + x)
Cross-multiplying: (100 + x) · (20 + x) = (50 + x) · (50 + x)
Multiplying out: 2000 + 120x + x2 = 2500 + 100x + x2
Simplifying: 20x = 500
x = 25
So, the numbers are 45, 75, and 125
The ratio: 125 / 75 = 5/3
and 75 / 45 = 5/3
