# Arithmetic and geometric question help

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#1
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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1 year ago
#2
(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 instead.

The first term of this geo sequence is , the second is , and the fourth is .

These form consecutive terms of an arithmetic sequence. So if we introduce some difference , then we start with and add to get . Hence . Also .

Subtract the first equation from the second, so you eliminate d, and you have .

a is nonzero hence you can cancel it, and hence solve the leftover cubic in r. The fact just means those are the values of r you're interested in, and there's only one of those when you solve the cubic.
1
#3
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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