Insert ‘n’ Geometric Means between Two Numbers a,b-product of n geometric means between a & b is equal to the nth power of single geometric mean between a & b

📢 Insert ‘n’ Geometric Means Between Two Numbers – Class 11 Mathematics 📚

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The concept of inserting ‘n’ Geometric Means (GM) between two numbers is a key topic in Class 11 Mathematics, particularly in Sequences & Series. This topic plays a crucial role in CBSE Board Exams, JEE Mains, and Advanced, helping students master Geometric Progression (GP) and its real-world applications.

Insert ‘n’ Geometric Means between Two Numbers a, b

If ‘n’ number of geometric means G1, G2, G3, G4, . . . , G_n are inserted between two numbers a and b, such that the resulting sequence a, G₁, G₂, G₃, G₄, . . ., G_n, b forms the geometric progression, then,

The first term = a and the (n + 2)th term = b

\(\begin{array}{l}i.e.\ b = a r ^{[ (n + 2) – 1 ]} = a r ^{(n + 1)}\end{array} \)

Or,

\(\begin{array}{l}\mathbf{\frac{b}{a}\;=\;r^{n\;+\;1}}\end{array} \)

Or,

\(\begin{array}{l}\mathbf{r\;=\;\left ( \;\frac{b}{a} \;\right )^{\frac{1}{n\;+\;1}}}\end{array} \)

Therefore,

\(\begin{array}{l}\mathbf{G_1 = \;a\;\left ( \;\frac{b}{a}\; \right )^{\frac{1}{n\;+\;1}}}\end{array} \)

Similarly,

\(\begin{array}{l}\mathbf{G_2 = \;a\;\left ( \;\frac{b}{a}\; \right )^{\frac{2}{n\;+\;1}}}\end{array} \)

Therefore, 

\(\begin{array}{l}\mathbf{G_n = \;a\;\left ( \;\frac{b}{a}\; \right )^{\frac{n}{n\;+\;1}}}\end{array} \)

Prove that product of n geometric means between a and b is equal to the nth power of the single geometric mean between a and b

Also, the product of n geometric means between a and b is equal to the nth power of the single geometric mean between a and b.

\(\begin{array}{l}i.e.\ \mathbf{\sum_{r\;=\;1}^{n}\;\;G_{r}\;=\;G^{n}}\end{array} \)

[where G = single geometric mean between a and b]

Or,

\(\begin{array}{l}G_{1} \times G_{2} \times G_{3} \times G_{4} \times . . . . . . . . . \times G_{n} = G^{n}\end{array} \)

🔍 Applications of Geometric Means

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

Solved Examples

📖 Example.1: Insert 3 Geometric Means Between 2 and 162

We need to insert 3 geometric means between 2 and 162.

Step 1: Identify given values

• First term: a = 2
• Last term: b = 162
• Number of GMs to insert: n = 3

Step 2: Find the common ratio rrr

Using the formula:

r=(b/a)^1/n+1

​ r=(162/2)^1/4

r=(81)^1/4

​r=3

Step 3: Find the inserted geometric means

GM₁=2×3=6

GM₂=6×3=18

GM₃​=18×3=54

Thus, the sequence is: 2, 6, 18, 54, 162 ✅

📖 Example 2: Insert three geometric means between 2 and 162.

Solution:

Given a = 2, b = 162, and n = 3

Therefore,

\(\begin{array}{l}\mathbf{r\;=\;\left ( \frac{b}{a} \right )^{\frac{1}{n\;+\;1}}\;=\;\left ( \frac{162}{2} \right )^{\frac{1}{4}}\;=\;\left ( 81 \right )^{\frac{1}{4}}\;=\;3}\end{array} \)

Hence, the 3 geometric means between 2 and 162 are 6, 18, and 54.

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📕 Author: Neeraj Anand
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