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# Algebra

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When the same constant is added to the numbers $a,$ $b,$ and $c,$ a three-term geometric sequence arises. If $a = 60,$ $b = 100,$ and $c = 140,$ what is the common ratio of the resulting sequence?

Aug 17, 2024

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First, we can write a handy equation to solve for the constant and find the 3 terms.

Let's let the constant added to every number be x.

Since the three terms for a geometric series, we have the equation

$$\frac{100+x}{60+x} = \frac{140+x}{100+x}$$

Now, when we crossmultiply and then expand everything out, we get

$$(x+100)(100+x)=(x+60)(140+x)\\ 200x+x^{2}+10000=140x+x^{2}+{8400+60x}\\ 200x+x^{2}+10000=200x+x^{2}+8400$$

Now, we bring all terms to one side of the equation. We have

$$200x+x^{2}+10000-x^{2}-200x-8400=0\\200x+1600-200x=0 \\ 1600=0$$

However, this statement is obviously not true, meaning that x is invalid.

This also means that there are NO solutions to this given problem.

*Note, I may have made a mistake. Not sure.

Thanks! :)

Aug 17, 2024
edited by NotThatSmart  Aug 17, 2024