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Q&A

What's the common ratio for this geometric sequence?

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Delayed learning some math, now I'm back at it.

Geometric sequences. Basically a sequence where it has a common ratio.

An example:

Sequence = {14, 28, 56, 112}
Ratio = 2
Proof = 14 * 2 * 2 * 2...


You can see where I'm going with this.

In my previous sequence question, I asked how I'm supposed to get the arithmetic means between two values. Here, I'm tasked to get geometric means this time. Getting geometric means is a lot more complicated because it requires multiplying and dividing instead.

So what are the values at the ends of the sequence? $16$ and $81$, and I have to find $3$ geometric means between them. My problem though is that I can't really use the same way I did with arithmetic sequences, so finding a common ratio is difficult.

Sequence = {16, ?, ?, ?, 81}
Ratio = ?
Proof = 16 * ? * ? * ? * ?...


Any ways of obtaining the common ratio of the sequence?

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If you look at the first sequence and divide every term by 14, you get a much simpler sequence. Consider how you could find the ratio from this and try the method on the second sequence.

Further solution/hint

You can divide by the first term in the second sequence, too.

What remains is essentially a problem of type $$r^4 = 81/16,$$ where $r$ is the unknown ratio.

Finally

From the previous we get $r = 3/2$. Repeated multiplication verifies that this is correct.

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