- A sequence is an ordered list of numbers or objects following a rule; each entry is a term.
- Inspect successive differences, ratios, positions and shapes. Several rules can fit a short finite list, so a pattern must be specified or justified.
- An AP has constant difference d between successive terms.
- For first term a and common difference d, a_n=a+(n−1)d.
- Use term position and the AP formula to solve for a, d or n.
- S_n=n/2[2a+(n−1)d]=n(a+l)/2 if l is the last term.
- The arithmetic mean of a and b is (a+b)/2 and lies midway between them.
- A GP has constant nonzero ratio r between consecutive terms when defined.
- For first term a and ratio r, a_n=ar^(n−1).
- For r≠1, S_n=a(r^n−1)/(r−1); for r=1, S_n=na.
- Square numbers are n²; triangular numbers are n(n+1)/2.
- A recurrence defines each term using earlier terms and starting values.
Use these notes alongside your current textbook. Check important formulas, worked answers and syllabus coverage with your teacher or official learning material.
Predicting What Comes Next: Exploring Sequences and Progressions — Class 9 Mathematics
Textbook edition: Ganita Manjari 2026–27, Part I, Chapter 8. This belongs to the 2026–27 Ganita Manjari chapter sequence.
Summary: A sequence is an ordered list of numbers or objects following a rule; each entry is a term.
These are original detailed supplementary notes and practice sets. Exact official exercise numbering, textbook diagrams and activity-by-activity solutions have not been verified against the full prescribed NCERT PDF.
Learning outcomes
- Define terms and state the conditions for formulas and theorems.
- Follow mathematical arguments step by step.
- Apply concepts to new examples and check the results.
- Identify common mistakes and invalid assumptions.
Sequences and terms
A sequence is an ordered list of numbers or objects following a rule; each entry is a term.
Worked example
2,5,8,11,... has first term 2 and fourth term 11.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Recognising patterns
Inspect successive differences, ratios, positions and shapes. Several rules can fit a short finite list, so a pattern must be specified or justified.
Worked example
2,4,8 could continue 16 by doubling, but three terms alone do not prove that rule.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Arithmetic progression
An AP has constant difference d between successive terms.
Worked example
7,11,15,19,... has d=4.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Nth term of an AP
For first term a and common difference d, a_n=a+(n−1)d.
Worked example
For a=3,d=5, a_8=3+7×5=38.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Finding an unknown term
Use term position and the AP formula to solve for a, d or n.
Worked example
In 4,9,14,..., 49 is the 10th term because 4+9×5=49.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Sum of first n AP terms
S_n=n/2[2a+(n−1)d]=n(a+l)/2 if l is the last term.
Worked example
1+2+...+20=20×21/2=210.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Arithmetic mean
The arithmetic mean of a and b is (a+b)/2 and lies midway between them.
Worked example
Between 6 and 14, arithmetic mean=10.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Geometric progression
A GP has constant nonzero ratio r between consecutive terms when defined.
Worked example
3,6,12,24,... has r=2.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Nth term of a GP
For first term a and ratio r, a_n=ar^(n−1).
Worked example
For a=2,r=3, a_5=2×3⁴=162.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Sum of a finite GP
For r≠1, S_n=a(r^n−1)/(r−1); for r=1, S_n=na.
Worked example
1+2+4+8+16=31.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Square and triangular numbers
Square numbers are n²; triangular numbers are n(n+1)/2.
Worked example
Sixth triangular number=6×7/2=21.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Recursive rules
A recurrence defines each term using earlier terms and starting values.
Worked example
Fibonacci-type sequence 1,1,2,3,5,8,... adds previous two.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Visual patterns
Represent growing figures with dot diagrams, tables or formulas to generalise the nth stage.
Worked example
A row of n connected squares uses 3n+1 matchsticks if adjacent squares share sides.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Real-world sequences
Regular savings increments may form an AP; repeated multiplication may form a GP. Models require explicit assumptions.
Worked example
₹100,₹120,₹140,... is AP with d=₹20.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Checking a prediction
Verify proposed formulas against all supplied terms and state why they apply.
Worked example
a_n=2n+1 gives 3,5,7,9 for n=1,2,3,4.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Common errors
AP difference and GP ratio are not interchangeable; in nth-term formulas count n−1 steps from first term.
Worked example
For 5,8,11,..., a_4=5+3×3=14.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Concepts and formulas at a glance
| Topic | Explanation | Illustration |
|---|---|---|
| Sequences and terms | A sequence is an ordered list of numbers or objects following a rule; each entry is a term. | 2,5,8,11,... has first term 2 and fourth term 11. |
| Recognising patterns | Inspect successive differences, ratios, positions and shapes. Several rules can fit a short finite list, so a pattern must be specified or justified. | 2,4,8 could continue 16 by doubling, but three terms alone do not prove that rule. |
| Arithmetic progression | An AP has constant difference d between successive terms. | 7,11,15,19,... has d=4. |
| Nth term of an AP | For first term a and common difference d, a_n=a+(n−1)d. | For a=3,d=5, a_8=3+7×5=38. |
| Finding an unknown term | Use term position and the AP formula to solve for a, d or n. | In 4,9,14,..., 49 is the 10th term because 4+9×5=49. |
| Sum of first n AP terms | S_n=n/2[2a+(n−1)d]=n(a+l)/2 if l is the last term. | 1+2+...+20=20×21/2=210. |
| Arithmetic mean | The arithmetic mean of a and b is (a+b)/2 and lies midway between them. | Between 6 and 14, arithmetic mean=10. |
| Geometric progression | A GP has constant nonzero ratio r between consecutive terms when defined. | 3,6,12,24,... has r=2. |
| Nth term of a GP | For first term a and ratio r, a_n=ar^(n−1). | For a=2,r=3, a_5=2×3⁴=162. |
| Sum of a finite GP | For r≠1, S_n=a(r^n−1)/(r−1); for r=1, S_n=na. | 1+2+4+8+16=31. |
| Square and triangular numbers | Square numbers are n²; triangular numbers are n(n+1)/2. | Sixth triangular number=6×7/2=21. |
| Recursive rules | A recurrence defines each term using earlier terms and starting values. | Fibonacci-type sequence 1,1,2,3,5,8,... adds previous two. |
| Visual patterns | Represent growing figures with dot diagrams, tables or formulas to generalise the nth stage. | A row of n connected squares uses 3n+1 matchsticks if adjacent squares share sides. |
| Real-world sequences | Regular savings increments may form an AP; repeated multiplication may form a GP. Models require explicit assumptions. | ₹100,₹120,₹140,... is AP with d=₹20. |
| Checking a prediction | Verify proposed formulas against all supplied terms and state why they apply. | a_n=2n+1 gives 3,5,7,9 for n=1,2,3,4. |
| Common errors | AP difference and GP ratio are not interchangeable; in nth-term formulas count n−1 steps from first term. | For 5,8,11,..., a_4=5+3×3=14. |
Practice set A — Definitions and foundations
- Define sequences and terms and create a correct example that satisfies all necessary conditions.
- Define recognising patterns and create a correct example that satisfies all necessary conditions.
- Define arithmetic progression and create a correct example that satisfies all necessary conditions.
- Define nth term of an ap and create a correct example that satisfies all necessary conditions.
- Define finding an unknown term and create a correct example that satisfies all necessary conditions.
- Define sum of first n ap terms and create a correct example that satisfies all necessary conditions.
- Define arithmetic mean and create a correct example that satisfies all necessary conditions.
- Define geometric progression and create a correct example that satisfies all necessary conditions.
- Define nth term of a gp and create a correct example that satisfies all necessary conditions.
- Define sum of a finite gp and create a correct example that satisfies all necessary conditions.
- Define square and triangular numbers and create a correct example that satisfies all necessary conditions.
- Define recursive rules and create a correct example that satisfies all necessary conditions.
Practice set B — Applications
- Verify this worked example: 2,5,8,11,... has first term 2 and fourth term 11. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: 2,4,8 could continue 16 by doubling, but three terms alone do not prove that rule. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: 7,11,15,19,... has d=4. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: For a=3,d=5, a_8=3+7×5=38. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: In 4,9,14,..., 49 is the 10th term because 4+9×5=49. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: 1+2+...+20=20×21/2=210. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: Between 6 and 14, arithmetic mean=10. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: 3,6,12,24,... has r=2. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: For a=2,r=3, a_5=2×3⁴=162. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: 1+2+4+8+16=31. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: Sixth triangular number=6×7/2=21. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: Fibonacci-type sequence 1,1,2,3,5,8,... adds previous two. Change the numerical values or premises and solve the new problem with complete reasoning.
Practice set C — Higher-order reasoning
- Explain a central theorem or formula and its assumptions.
- Find a counterexample to an incorrect generalisation.
- Compare two concepts and explain their differences.
- Write a real-world application with relevant units or conditions.
- Identify and correct a plausible error in a solution.
- Verify a result using an alternative approach.
- Create an original multi-step problem and solve it.
- Explain the concept to a beginner using an accurate diagram or table.
Frequently asked questions with answers
Explain sequences and terms with an example.
A sequence is an ordered list of numbers or objects following a rule; each entry is a term. 2,5,8,11,... has first term 2 and fourth term 11.
Explain recognising patterns with an example.
Inspect successive differences, ratios, positions and shapes. Several rules can fit a short finite list, so a pattern must be specified or justified. 2,4,8 could continue 16 by doubling, but three terms alone do not prove that rule.
Explain arithmetic progression with an example.
An AP has constant difference d between successive terms. 7,11,15,19,... has d=4.
Explain nth term of an ap with an example.
For first term a and common difference d, a_n=a+(n−1)d. For a=3,d=5, a_8=3+7×5=38.
Explain finding an unknown term with an example.
Use term position and the AP formula to solve for a, d or n. In 4,9,14,..., 49 is the 10th term because 4+9×5=49.
Explain sum of first n ap terms with an example.
S_n=n/2[2a+(n−1)d]=n(a+l)/2 if l is the last term. 1+2+...+20=20×21/2=210.
Explain arithmetic mean with an example.
The arithmetic mean of a and b is (a+b)/2 and lies midway between them. Between 6 and 14, arithmetic mean=10.
Explain geometric progression with an example.
A GP has constant nonzero ratio r between consecutive terms when defined. 3,6,12,24,... has r=2.
Explain nth term of a gp with an example.
For first term a and ratio r, a_n=ar^(n−1). For a=2,r=3, a_5=2×3⁴=162.
Explain sum of a finite gp with an example.
For r≠1, S_n=a(r^n−1)/(r−1); for r=1, S_n=na. 1+2+4+8+16=31.
Explain square and triangular numbers with an example.
Square numbers are n²; triangular numbers are n(n+1)/2. Sixth triangular number=6×7/2=21.
Explain recursive rules with an example.
A recurrence defines each term using earlier terms and starting values. Fibonacci-type sequence 1,1,2,3,5,8,... adds previous two.
Explain visual patterns with an example.
Represent growing figures with dot diagrams, tables or formulas to generalise the nth stage. A row of n connected squares uses 3n+1 matchsticks if adjacent squares share sides.
Explain real-world sequences with an example.
Regular savings increments may form an AP; repeated multiplication may form a GP. Models require explicit assumptions. ₹100,₹120,₹140,... is AP with d=₹20.
Explain checking a prediction with an example.
Verify proposed formulas against all supplied terms and state why they apply. a_n=2n+1 gives 3,5,7,9 for n=1,2,3,4.
Revision checklist
- I can explain and apply sequences and terms.
- I can explain and apply recognising patterns.
- I can explain and apply arithmetic progression.
- I can explain and apply nth term of an ap.
- I can explain and apply finding an unknown term.
- I can explain and apply sum of first n ap terms.
- I can explain and apply arithmetic mean.
- I can explain and apply geometric progression.
- I can explain and apply nth term of a gp.
- I can explain and apply sum of a finite gp.
- I can explain and apply square and triangular numbers.
- I can explain and apply recursive rules.
- I can explain and apply visual patterns.
- I can explain and apply real-world sequences.
- I can explain and apply checking a prediction.
- I can explain and apply common errors.
Official textbook and accuracy note
Official NCERT textbook portal. Supplementary original study guide and practice exercises, not a verified reproduction of numbered NCERT exercises or official answer key.
Frequently asked questions
Explain sequences and terms with an example.
A sequence is an ordered list of numbers or objects following a rule; each entry is a term. 2,5,8,11,... has first term 2 and fourth term 11.
Explain recognising patterns with an example.
Inspect successive differences, ratios, positions and shapes. Several rules can fit a short finite list, so a pattern must be specified or justified. 2,4,8 could continue 16 by doubling, but three terms alone do not prove that rule.
Explain arithmetic progression with an example.
An AP has constant difference d between successive terms. 7,11,15,19,... has d=4.
Explain nth term of an ap with an example.
For first term a and common difference d, a_n=a+(n−1)d. For a=3,d=5, a_8=3+7×5=38.
Explain finding an unknown term with an example.
Use term position and the AP formula to solve for a, d or n. In 4,9,14,..., 49 is the 10th term because 4+9×5=49.
Explain sum of first n ap terms with an example.
S_n=n/2[2a+(n−1)d]=n(a+l)/2 if l is the last term. 1+2+...+20=20×21/2=210.
Explain arithmetic mean with an example.
The arithmetic mean of a and b is (a+b)/2 and lies midway between them. Between 6 and 14, arithmetic mean=10.
Explain geometric progression with an example.
A GP has constant nonzero ratio r between consecutive terms when defined. 3,6,12,24,... has r=2.
Explain nth term of a gp with an example.
For first term a and ratio r, a_n=ar^(n−1). For a=2,r=3, a_5=2×3⁴=162.
Explain sum of a finite gp with an example.
For r≠1, S_n=a(r^n−1)/(r−1); for r=1, S_n=na. 1+2+4+8+16=31.
Explain square and triangular numbers with an example.
Square numbers are n²; triangular numbers are n(n+1)/2. Sixth triangular number=6×7/2=21.
Explain recursive rules with an example.
A recurrence defines each term using earlier terms and starting values. Fibonacci-type sequence 1,1,2,3,5,8,... adds previous two.
Explain visual patterns with an example.
Represent growing figures with dot diagrams, tables or formulas to generalise the nth stage. A row of n connected squares uses 3n+1 matchsticks if adjacent squares share sides.
Explain real-world sequences with an example.
Regular savings increments may form an AP; repeated multiplication may form a GP. Models require explicit assumptions. ₹100,₹120,₹140,... is AP with d=₹20.
Explain checking a prediction with an example.
Verify proposed formulas against all supplied terms and state why they apply. a_n=2n+1 gives 3,5,7,9 for n=1,2,3,4.
