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# sequences

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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?

$$\phantom{60, 100, and 160}$$

Jun 30, 2022

#1
+1

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 $$8 0\div 40 = \color{brown}\boxed2$$

Jun 30, 2022

#1
+1

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 $$8 0\div 40 = \color{brown}\boxed2$$

BuilderBoi Jun 30, 2022