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, √(4×7) = √28 = 2√7.
Formula
GM = Antilog (∑ log xk)/n
where,
∑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 = √(x1 × x2 × x3 × … × xn)
GM = (x1 × x2 × x3 × … × xn)1/n
Taking log on both sides of the equation we get,
log GM = log (x1 × x2 × x3 × … × xn)1/n
Using the property log ab = b log a, we get
log GM = (1/n) log (x1 × x2 × x3 × … × 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 = (∑ log xk)/n
Taking antilog on both sides we get,
GM = Antilog (∑ 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 × 4 × 6 × 8 × 10 × 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 × 8 × 12 × 16 × 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 × 10 × 15 × 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