What's the common ratio for this geometric sequence?
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?
1 answer
The following users marked this post as Works for me:
| User | Comment | Date |
|---|---|---|
| General Sebast1an | (no comment) | Jul 29, 2022 at 00:49 |
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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