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# Sequences

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Suppose w,x,y,z are positive real numbers such that w,x,y,z form an increasing arithmetic sequence, and w,x,z form a geometric sequence.

What is the value of w/z ? The answer must be in fraction, dont divide it.

Thank you!!!

May 10, 2022

#1
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With a bit of trial and error, we can find the sequence to be 2,4,6,8.

Can you take it from here?

May 10, 2022
#2
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I kind of understandbut could you please deeply elaborate on how to get the answer since I have multiple question like that so if you ahve a method I could find it useful.

May 10, 2022
#3
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Sorry about that, hopefully, this will make it easier for you!!

Let x be the first term and a be a common difference.

We have the arithmetic sequence: $$x, x+a, x+ 2a, x+ 3a$$ and the geometric sequence is: $$x, x+a, x+ 3a$$

From the geometric sequence, we can derive the equation: $$\large{{{x +a} \over x} ={ {x+3a} \over {x+a}}}$$

Basically, this equation states that the common difference between the first 2 terms (x+a and x) is the same as the common difference between the next 2 terms (x+3a and x+a).

From the geometric series, we know that $${{x+3a}\over x} = 2a$$ (value of 3rd term divided by 1st term is 2 times the common difference)

When we solve this system, we find that $$x = a = 2$$, meaning the series is 2,4,6,8.

BuilderBoi  May 10, 2022