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?

ABJeIIy Jun 16, 2024

#1**+1 **

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! :)

NotThatSmart Jun 16, 2024