Geometric Mean(GM)-Definition, Formulas, Solved Examples & Properties, FAQs, Practice Questions

📢 Master Geometric Mean (GM) – Class 11 Mathematics 📚

🚀 Ace Board Exams & JEE Mains/Advanced with Neeraj Anand’s Expert Guide! 🚀

The Geometric Mean (GM) is a fundamental concept in Class 11 Mathematics, widely used in Sequences & Series, Growth Calculations, Financial Mathematics, and Scientific Applications. It plays a crucial role in Board Exams and JEE Mains/Advanced by helping students solve complex problems efficiently.

🔢 What is Geometric Mean (GM)?

The Geometric Mean (GM) between two or more numbers is the central value of a set of numbers in a geometric sequence. Unlike the Arithmetic Mean (AM), which is based on addition, the GM is based on multiplication and roots.

Geometric mean is defined as the nth root of the product of n numbers. If the terms a, b, and c are in geometric progression (GP), then the middle term (b) is called the geometric mean (GM) of the other two terms (a and c). Therefore, the geometric mean between two terms, a and c, is given by

GM (b) = √(ac)

(Where, a and c > 0)

The geometric mean of as sequence a1, a2, a3, a4, a5,…. up to n terms can be defined as

\(\begin{array}{l}\mathbf{GM\;=\;\left ( \;a_{1}\;\times \;a_{2}\;\times\; a_{3}\;\times \;a_{4}\;\times \;a_{n} \;\right )^{\frac{1}{n}}}\end{array} \)

Geometric Mean Formula

The formula to calculate the geometric mean is given below:

The Geometric Mean (G.M) of a series containing n observations is the nth root of the product of the values.

Consider, if x₁, x₂ …. X_n are the observation, then the G.M is defined as:

\(\begin{array}{l}G. M = \sqrt[n]{x_{1}\times x_{2}\times …x_{n}}\end{array} \) or \(\begin{array}{l}G. M = (x_{1}\times x_{2}\times …x_{n})^{^{\frac{1}{n}}}\end{array} \)

This can also be written as;

\(\begin{array}{l}Log\ GM =\frac{1}{n}\log (x_{1}\times x_{2}\times ….x_{n})\end{array} \)

\(\begin{array}{l}=\frac{1}{n}(\log x_{1}+\log x_{2}+….+\log x_{n})\end{array} \)

\(\begin{array}{l}=\frac{\sum \log x_{i}}{n}\end{array} \)

Therefore, Geometric Mean,

\(\begin{array}{l}GM = Antilog\frac{\sum \log x_{i}}{n}\end{array} \)

Where n = f₁ + f₂ +…..+ f_n

It is also represented as:

\(\begin{array}{l}G.M. =\sqrt[n]{\prod_{i=1}^{n}x_{i}}\end{array} \)

For any Grouped Data, G.M can be written as;

\(\begin{array}{l}GM = Antilog\frac{\sum f \log x_{i}}{n}\end{array} \)

Suppose we are given 3 numbers 3, 9, and 27 then the geometric mean of the given values is calculated by taking the third root of the product of the three given data. The calculation of the Geometric Mean is shown below:

∛(3×9×27) = ∛(729) = 9

Difference Between Arithmetic Mean and Geometric Mean

Arithmetic Mean	Geometric Mean
Arithmetic mean is the measure of the central tendency it is found by taking sum of all the values and then dividing it by the numbers of values.	Geometric mean is also the measure of the central tendency. It is calculating by first taking the product of all n value and then taking the n the roots of the values.
Arithmetic Mean Formula,AM = (Sum of Value)/(Number of Values)AM = (x1 + x2 + … + xn)/n	Geometric Mean Formula,GM = (x1 × x2 × … × xn)1/n
Example: Find the arithmetic mean of 4, 6, 10, 8Given values,4, 6, 10 and 8Number of Values = 4Sum of Value = 4+6+10+8 = 28AM = 28/4 = 7	Example: Find the geometric mean of 4, 6, 10, 8Given values,4, 6, 10 and 8Number of Values = 4Product of Value = 4×6×10×8 = 1920GM = (1920)1/4 = 6.2

How to Find the Geometric Mean?

Here are the simple steps to find the geometric mean of a set of numbers:

• Step 1: Multiply all the numbers together.
• Step 2: Count the total number of values (n).
• Step 3: Take the n-th root of the product.

The geometric mean of two numbers is found using the geometric mean formula, GM = √(ab), where a and b are the two numbers.

Geometric Mean Properties

Some of the important properties of the G.M are:

• The G.M for the given data set is always less than the arithmetic mean for the data set
• If each object in the data set is substituted by the G.M, then the product of the objects remains unchanged.
• The ratio of the corresponding observations of the G.M in two series is equal to the ratio of their geometric means
• The products of the corresponding items of the G.M in two series are equal to the product of their geometric mean.

📘 Applications of Geometric Mean in Real Life

📌 Finance & Economics – Compound Interest, Stock Market Growth
📌 Physics – Harmonic Motion, Sound Waves
📌 Biology – Bacterial Growth, Population Studies
📌 Engineering – Signal Processing, Electrical Circuits
📌 Statistics – Data Analysis, Predictive Modeling

Geometric Mean Solved Examples

Example.1: What is the geometric mean of 36 and 4?

Solution:

Let the geometric mean of 36 and 4 is g,

g = √(36.4) = √(144)

g = 12

Thus, the geometric mean of 36 and 4 is 12.

Example.2 : Find the geometric mean of the numbers 2, 4, 8, and 16.

Solution

Given numbers: 2, 4, 8, 16

n = 4

Multiply the numbers together:
2 × 4 × 8 × 16 = 1024.

Take the 4th root (since there are 4 numbers):
∜1024 = 5.5 (approximately).

The geometric mean of 2, 4, 8, and 16 is approximately 5.5.

Example.3: Calculate the geometric mean of the sequence, 2, 4, 6, 8, 10, 12.

Solution:

Given,

• 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,

GM = (P)^1/n

GM = (46080)^1/6

GM = 5.98

Example 4: Calculate the geometric mean of the sequence, 4, 8, 12, 16, 20.

Solution:

Given,

• Sequence, 4, 8, 12, 16, 20

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

Number of terms (n) = 5

Using the formula,

GM = (P)^1/n

GM = (122880)1/5

GM = 10.42

Example 5: Calculate the geometric mean of the sequence, 5, 10, 15, 20.

Solution:

Given,

• Sequence, 5, 10, 15, 20

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

Number of terms (n) = 4

Using the formula,

GM = (P)^1/n

GM = (15000)^1/4

GM = 11.06

Example 6: Find the number of terms in a sequence if the geometric mean is 32 and the product of terms is 1024.

Solution:

Given,

• Product of terms (P) = 1024
• GM of terms = 32

Using the formula,

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

Example 7: Find the number of terms in a sequence if the geometric mean is 8 and the product of terms is 4096.

Solution:

Given,

• Product of terms (P) = 4096
• GM of terms = 8

Using the formula,

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

Example 8: Find the number of terms in a sequence if the geometric mean is 4 and the product of terms is 65536.

Solution:

Given,

• Product of terms (P) = 65536
• GM of terms = 4

Using the formula,

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

Example 9: Find the number of terms in a sequence if the geometric mean is 16 and the product of terms is 16777216.

Solution:

Given,

• 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

Practice Questions on Geometric Mean

Q1. Calculate the geometric mean of the sequence, 15, 25, 35, 45.

Q2. What is the geometric mean of 7 and 28?

Q3. Find the number of terms in a sequence if the geometric mean is 22 and the product of terms is 655360.

Q4. What is the geometric mean of 4 and 25?

Q5. A company has seen its yearly revenue grow by 10%, 15%, and 20% over the past three years. What is the geometric mean of the growth rate over these three years?

Q6. A gardener is planting three types of flowers with heights measured in centimeters as follows: 12 cm, 18 cm and 30 cm. Calculate the geometric mean of the heights of the flowers.

Q7. A researcher measured the weights (in kg) of five different fruits: 0.5, 0.75, 1.0, 1.5 and 2.0. What is the geometric mean of these weights?

Q8. If the geometric mean of three numbers is 10 and one of the numbers is 5, what is the product of the other two numbers?

Q9. A certain stock has returns of 8%, 12% and 20% over three consecutive years. What is the geometric mean of the stock’s returns?

Q10. The prices of three products are $20, $50 and $80. Calculate the geometric mean of the prices to determine the average price level.

Answer Key:

1. 25.97
2. 14
3. 4
4. 10
5. 12.16%
6. 18.83 cm
7. 0.89 kg
8. 50
9. 13.86%
10. $40.00

Geometric Mean – FAQs

What is Geometric Mean?

Geometric mean of numbers is time measure of the central tendency that is used to find the central values of the given data set. It is found by taking the product of all the given value and then taking the nth roots of the number.

What is Formula for Geometric Mean?

The formula to calculate the geometric mean is added below, suppose we gave n numbers x₁, x₂, …. x_n then the geometric mean formula is,

GM = (x₁ × x₂ × … × x_n)1/n

What is AM, GM Inequality?

The AM and GM inequality is the inequality that states that, AM is always greater than equal to the GM. This is represented as,

A.M ≥ G.M

What is Relation Between AM, GM and HM?

For given n numbers the relation between AM, GM and HM is,

GM² = AM × HM

What is Geometric Mean in Statistics?

The geometric mean in statistics is the average multiple of all the value of the given numbers. Geometric mean is found by taking the multiple of all the number and then taking the n th root of the number.

What is the Geometric Mean of 4 and 6?

The geometric mean of 4 and 6 is 4.89 (approx).

What is the Geometric Mean of 4 and 66?

The geometric mean of 4 and 16 is 8.

What is the Geometric Mean of 9 and 4?

The geometric mean of 9 and 4 is 6.

Why Geometric Mean is Better than Arithmetic Mean?

Both the geometric mean and arithmetic mean are used to determine the average. For any two positive unequal numbers, the geometric mean is always less than the arithmetic mean. Now, the geometric mean is better since it takes indicates the central tendency. In certain cases, arithmetic mean works better like in representing average temperatures, etc.

📚 Why Study from Neeraj Anand’s Book?

✔️ Conceptual Clarity – Easy-to-understand explanations
✔️ Extensive Practice Questions – Includes solved examples & exercises
✔️ JEE & Board Exam Focused – Tricks and shortcuts for fast problem-solving

📥 Download the PDF & Start Learning Today!

📕 Author: Neeraj Anand
🏛 Publisher: ANAND TECHNICAL PUBLISHERS
🏫 Available at: ANAND CLASSES

💯 Master Geometric Mean & Excel in Your CBSE Board & JEE Exams!