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# Geometric Mean Formula

In arithmetic, the geometric mean is defined as an average in which the numbers of a sequence are multiplied together and then its nth root is computed, where n is the number of values in that series. It can be interpreted as the nth root of the product of n values. By calculating the root of the product of their values, the geometric mean denotes the central tendency of a collection of numbers. For example, if we are given two numbers 4 and 7, the geometric mean is given by, âˆš(4Ã—7) = âˆš28 = 2âˆš7.

Formula

GM = Antilog (âˆ‘ log xk)/n

where,

âˆ‘log xk is the logarithm value of sum of all the values in a sequence,

n is the number of values in the sequence.

Derivation

Suppose x1, x2, x3, x4, ……, xn are the values of a sequence whose geometric mean has to be evaluated.

So, the geometric mean of the given sequence can be written as,

GM = âˆš(x1 Ã— x2 Ã— x3 Ã— … Ã— xn)

GM = (x1 Ã— x2 Ã— x3 Ã— … Ã— xn)1/n

Taking log on both sides of the equation we get,

log GM = log (x1 Ã— x2 Ã— x3 Ã— … Ã— xn)1/n

Using the property log ab = b log a, we get

log GM = (1/n) log (x1 Ã— x2 Ã— x3 Ã— … Ã— xn)

Using the property log (ab) = log a + log b, we get

log GM = (1/n) (log x1 + log x2 + log x3 + … + log xn)

log GM = (âˆ‘ log xk)/n

Taking antilog on both sides we get,

GM = Antilog (âˆ‘ log xk)/n

This derives the formula for geometric mean of a series.

### Sample Problems

Problem 1. Find the geometric mean of the sequence, 2, 4, 6, 8, 10, 12.

Solution:

We have the sequence, 2, 4, 6, 8, 10, 12.

Product of terms (P) = 2 Ã— 4 Ã— 6 Ã— 8 Ã— 10 Ã— 12 = 46080

Number of terms (n) = 6

Using the formula we have,

GM = (P)1/n

= (46080)1/6

= 5.98

Problem 2. Find the geometric mean of the sequence, 4, 8, 12, 16, 20.

Solution:

We have the sequence, 4, 8, 12, 16, 20.

Product of terms (P) = 4 Ã— 8 Ã— 12 Ã— 16 Ã— 20 = 122880

Number of terms (n) = 5

Using the formula we have,

GM = (P)1/n

= (122880)1/5

= 10.42

Problem 3. Find the geometric mean of the sequence, 5, 10, 15, 20.

Solution:

We have the sequence, 5, 10, 15, 20.

Product of terms (P) = 5 Ã— 10 Ã— 15 Ã— 20 = 15000

Number of terms (n) = 4

Using the formula we have,

GM = (P)1/n

= (15000)1/4

= 11.06

Problem 4. Find the number of terms in a sequence if the geometric mean is 32 and the product of terms is 1024.

Solution:

We have,

Product of terms (P) = 1024

GM of terms = 32

Using the formula we have,

GM = (P)1/n

=> 1/n = log GM/log P

=> n = log P/log GM

=> n = log 1024/log 32

=> n = 10/5

=> n = 2

Problem 5. Find the number of terms in a sequence if the geometric mean is 8 and the product of terms is 4096.

Solution:

We have,

Product of terms (P) = 4096

GM of terms = 8

Using the formula we have,

GM = (P)1/n

=> 1/n = log GM/log P

=> n = log P/log GM

=> n = log 4096/log 8

=> n = 12/3

=> n = 4

Problem 6. Find the number of terms in a sequence if the geometric mean is 4 and the product of terms is 65536.

Solution:

We have,

Product of terms (P) = 65536

GM of terms = 4

Using the formula we have,

GM = (P)1/n

=> 1/n = log GM/log P

=> n = log P/log GM

=> n = log 65536/log 4

=> n = 16/2

=> n = 8

Problem 7. Find the number of terms in a sequence if the geometric mean is 16 and the product of terms is 16777216.

Solution:

We have,

Product of terms (P) = 16777216

GM of terms = 16

Using the formula we have,

GM = (P)1/n

=> 1/n = log GM/log P

=> n = log P/log GM

=> n = log 16777216/log 16

=> n = 24/4

=> n = 6

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