Insert a geometric mean between k and 1/k
THE GEOMETRIC MEAN
The geometric mean between k and 1/k is 1.
What is a geometric sequence?
A sequence is a sequence of numbers that follows a pattern or rule. The rule of multiplying or dividing by a constant number (known as the common ratio) each time is called a geometric sequence.
The Common Ratio
A common ratio is a number that is multiplied or divided at each stage of a sequence. It is found by dividing pairs of two consecutive terms. It doesn’t matter which pair is chosen as long as they are adjacent to each other.
- 3, 9, 27, 81, …
In General, if the first term of the sequence is a and the common ratio is r, we could write a geometric sequence like this:
- a, ar, ar², ar³, …
A geometric sequence can be written as a rule:
- nth term = ar^(n-1)
- a_n = nth term
- a = the first term of the sequence
- r = the common ration
- Insert a geometric mean between k and 1/k.
Let the sequence is k, ar, 1/k.
- k, ar, 1/k = a, ar, ar²
Now we compare the given number to find ar
- a = k
- ar² = 1/k
Substitute a to ar²
- kr² = 1/k
Divide by k
- r² = 1/k²
- r = 1/k
We get r = 1/k.
Therefore, the geometric mean between k and 1/k is
- ar = k · 1/k = 1.
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