Arithmetic and geometric question help

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Officialbellew
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Hi, I’m really stuck on the question below if anyone could help that would be much appreciated.

“The first, second and fourth terms of a geometric sequence form consecutive terms of an arithmetic sequence.

Given that the sum to infinity of the geometric sequence exists, find the exact value of the common ratio.”

I’ve formed 4 equations from the information but I’m confused as to how to equate and solve the equations to find each unknown. Also I understand that |r|>1 because the sum to infinity exists but I’m not sure how I use this to solve for the common ratio. Any help would be appreciated.

Many thanks,
Belle
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RDKGames
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(Original post by Officialbellew)
Hi, I’m really stuck on the question below if anyone could help that would be much appreciated.

“The first, second and fourth terms of a geometric sequence form consecutive terms of an arithmetic sequence.

Given that the sum to infinity of the geometric sequence exists, find the exact value of the common ratio.”

I’ve formed 4 equations from the information but I’m confused as to how to equate and solve the equations to find each unknown. Also I understand that |r|>1 because the sum to infinity exists but I’m not sure how I use this to solve for the common ratio. Any help would be appreciated.

Many thanks,
Belle
Actually it's the other way round. Value for infinite sum exists only if |r|<1 instead.

The first term of this geo sequence is a, the second is ar, and the fourth is ar^3.

These form consecutive terms of an arithmetic sequence. So if we introduce some difference d, then we start with a and add d to get ar. Hence a+d=ar. Also ar+d=ar^3.

Subtract the first equation from the second, so you eliminate d, and you have ar-a = ar^3-ar.

a is nonzero hence you can cancel it, and hence solve the leftover cubic in r. The fact |r| < 1 just means those are the values of r you're interested in, and there's only one of those when you solve the cubic.
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Officialbellew
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Amazing thank you. I was on the right track and had the equations just got so consumed by the question that I got confused!! Thank you.
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