Arithmetic and geometric means: Class 11 practice

What you should know first

• AP and GP

The idea behind arithmetic and geometric means

The arithmetic mean of a and b is (a + b)/2; the geometric mean is √(ab) (for positive a, b). Inserting n means between a and b creates an AP (or GP) with n + 2 terms.

Step-by-step method

1. AM = (a + b)/2; GM = √(ab).
2. To insert n AMs: d = (b − a)/(n + 1).
3. List the terms.

Worked example 1 (Basic)

Find the AM and GM of 6 and 96.

1. AM = (6 + 96)/2 = 51
2. GM = √(576) = 24
3. Answer: AM 51, GM 24

Worked example 2 (Standard)

Insert 5 arithmetic means between 8 and 26.

1. d = (26 − 8)/6 = 3
2. Means: 11, 14, 17, 20, 23
3. Answer: 11, 14, 17, 20, 23

Worked example 3 (Challenge)

Insert 2 geometric means between 2 and 16.

1. r³ = 8 → r = 2
2. Means: 4, 8
3. Answer: 4, 8

Common mistakes and misconceptions

• Using n instead of n + 1 for the common difference
• Taking GM of negative numbers

Practice questions

1. (Basic) Find the AM and GM of 4 and 100.
2. (Basic) Find the AM and GM of 5 and 20.
3. (Standard) Insert 3 arithmetic means between 8 and 12.
4. (Standard) Insert 3 arithmetic means between 14 and 38.
5. (Challenge) Insert 2 geometric means between 2 and 54.
6. (Challenge) Insert 2 geometric means between 3 and 81.

Answer key with full working

• Q1: AM = (4 + 100)/2 = 52 → GM = √(400) = 20 → Answer: AM 52, GM 20
• Q2: AM = (5 + 20)/2 = 12.5 → GM = √(100) = 10 → Answer: AM 12.5, GM 10
• Q3: d = (12 − 8)/4 = 1 → Means: 9, 10, 11 → Answer: 9, 10, 11
• Q4: d = (38 − 14)/4 = 6 → Means: 20, 26, 32 → Answer: 20, 26, 32
• Q5: r³ = 27 → r = 3 → Means: 6, 18 → Answer: 6, 18
• Q6: r³ = 27 → r = 3 → Means: 9, 27 → Answer: 9, 27

Syllabus note

This skill is listed under "Sequences and Series" for Class 11 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

Related practice and notes

• Sequences and Series chapter notes (Class 11)
• Geometric progressions
• Permutations and combinations problems
• Binomial theorem: general term and coefficients
• Counting with Venn diagrams
• Domain and range of functions
• All Class 11 practice

Frequently asked questions

Which chapter is arithmetic and geometric means in?

"Sequences and Series", Class 11 Maths (NCERT/CBSE). Other boards may place it differently.

How were the answers checked?

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.