+1839 When the same constant is added to the numbers 60, 120, and 140, a three-term geometric sequence arises. What is the common ratio of the resulting sequence?
+1926 We can set variables to solve this problem.
Let's let x be the same constant added to every single number.
The 3 terms in the geometric series must be \(60+x, 120+x, 140+x\)
Since these 3 numbers form a geoemtric series, we can write the equation
\(\frac{60+x}{120+x}=\frac{120+x}{140+x}\)
Now, we simply solve for x.
We have \((60+x)(140+x)=(120+x)^2\)
Expanding out everything and combining all like terms, we get
\(-40x-6000=0\)
\(x=-150\)
Our geometric series is now \(-90, -30, -10\)
The common ratio is \(-30/-90 = 1/3\)
So our answer is 1/3.
Thanks! :)
