Insert a geometric mean between k and 1/k

Insert a geometric mean between k and 1/k

Answer :

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.

Example 2:

  • 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)

where:

  • a_n = nth term
  • a = the first term of the sequence
  • r = the common ration

GIVEN QUESTION

  • Insert a geometric mean between k and 1/k.

Solution

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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