Geometric Mean Examples

The Name ’Geometric’
If a, b, and c are any consecutive terms of a geometric sequence then b/a=c/b.
b2=ac and so b=±√ac where Mac is the geometric mean of a and c.

Geometric Mean

When three quantities are in G.P., the middle one is called the Geometric Mean (G.M.) between the other two. Thus G will be the G.M. between a and b if a, G, b are in G.P.

To Find G.M between a and b:
Let, G be the G.M. between a and b
Then a, G, b are in G.P. ∴

geometric mean a

Hence the G.M. between two quantities is equal to the square root of their product.

Example 1. Find the G.M. between 8 and 72.
Solution:

geometric mean b

Answer: G=±24

If we consider three consecutive terms in a geometric sequence {x, y, and z} then

geometric mean c

where r is the common factor.

Thus, the middle term, y, called the geometric mean, can be calculated in terms of the outer two terms, x and z.

For a geometric sequence {…, x, y, z, …}

y2=xz

If we need to find three unknown consecutive terms in geometric sequence, we let the terms be:
geometric mean d

Ex2. If 5, x, 45 are the first three terms of a geometric sequence, determine the value of x.

geometric mean e

Ex3. Find k given that 4, k, and k2-1 are consecutive terms of a geometric sequence.

Since the first two terms are 4 and k, the common ratio must be ¼k, the second term divided by the first one. That’s how you find a common ratio. This indicates that the ratio of the third term to the second must share that value. Not only does this give an equation to solve, it’s a proportion! Those are almost fun.

geometric mean f

Yes, there’s no reason that k can’t be negative. Feel free to check it if you’d like by writing down the terms. I just thought it through, and I’m convinced. And I don’t want to type any more fractions or radicals for this problem.
Let’s read post 👉Arithmetic Mean ≥ Geometric Mean ≥ Harmonic Mean (AM ≥ GM ≥ HM)👈.

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