When the same constant is added to the numbers 60, 100, and 180, a three-term geometric sequence arises. What is the common ratio of the resulting sequence?
\(\phantom{60, 100, and 160}\)
+2666 Let the constant be \(c\), and let the common difference be \(d\).
We have the equation from the first and second terms: \(d(60 + c) = 100 + c\), and likewise, we have \(d(100 + c) = 180 + c\)
Solving for \(d\) in both equations gives \(d = {100 + c \over 60 + c} = {180 + c \over 100+c}\)
Cross multiplying gives us: \(c^2 + 200c + 10000 = c^2+240c+10800\)
Solving, we find \(c =-20\), meaning the series is 40, 80, 160. The common ratio is \(8 0\div 40 = \color{brown}\boxed2\)
+2666 Let the constant be \(c\), and let the common difference be \(d\).
We have the equation from the first and second terms: \(d(60 + c) = 100 + c\), and likewise, we have \(d(100 + c) = 180 + c\)
Solving for \(d\) in both equations gives \(d = {100 + c \over 60 + c} = {180 + c \over 100+c}\)
Cross multiplying gives us: \(c^2 + 200c + 10000 = c^2+240c+10800\)
Solving, we find \(c =-20\), meaning the series is 40, 80, 160. The common ratio is \(8 0\div 40 = \color{brown}\boxed2\)
