Number Sequences (Grade 5)
A number sequence is an ordered list of numbers that follows a specific pattern or rule. Each number in the sequence is called a term. The challenge is to identify the rule and use it to find missing terms or extend the sequence.
In Class 5, you will work with arithmetic sequences (constant difference), geometric patterns (constant multiplier), and other patterns involving squares, cubes, or alternating operations. Recognising sequences builds algebraic thinking and prepares you for higher mathematics.
Sequences are everywhere: page numbers in a book, house numbers on a street, and the pattern of petals on flowers all follow sequences.
What is Number Sequences - Class 5 Maths (Patterns and Algebra)?
A number sequence is a list of numbers arranged in a definite order according to a rule.
- Term: Each number in the sequence.
- Rule: The pattern that connects one term to the next.
- Common difference: In an arithmetic sequence, the constant value added (or subtracted) to get the next term.
- Common ratio: In a geometric sequence, the constant value multiplied to get the next term.
Types and Properties
Types of number sequences:
- Arithmetic (add/subtract a constant): 3, 7, 11, 15, 19, ... (rule: add 4).
- Geometric (multiply/divide by a constant): 2, 6, 18, 54, ... (rule: multiply by 3).
- Square numbers: 1, 4, 9, 16, 25, ... (1², 2², 3², 4², 5²).
- Triangular numbers: 1, 3, 6, 10, 15, ... (add 2, add 3, add 4, add 5).
- Fibonacci-type: Each term is the sum of the two before it. 1, 1, 2, 3, 5, 8, 13, ...
- Alternating patterns: The rule switches between two operations.
Solved Examples
Example 1: Example 1: Arithmetic Sequence (Adding)
Problem: Find the next three terms: 5, 11, 17, 23, _____, _____, _____.
Solution:
Step 1: Find the difference: 11 − 5 = 6, 17 − 11 = 6, 23 − 17 = 6.
Step 2: Rule: add 6.
Step 3: Next terms: 23 + 6 = 29, 29 + 6 = 35, 35 + 6 = 41.
Answer: 29, 35, 41.
Example 2: Example 2: Arithmetic Sequence (Subtracting)
Problem: Find the next three terms: 50, 43, 36, 29, _____, _____, _____.
Solution:
Step 1: Difference: 43 − 50 = −7 (subtract 7 each time).
Step 2: Next terms: 29 − 7 = 22, 22 − 7 = 15, 15 − 7 = 8.
Answer: 22, 15, 8.
Example 3: Example 3: Geometric Sequence
Problem: Find the next two terms: 3, 9, 27, 81, _____, _____.
Solution:
Step 1: 9 ÷ 3 = 3, 27 ÷ 9 = 3, 81 ÷ 27 = 3.
Step 2: Rule: multiply by 3.
Step 3: 81 × 3 = 243, 243 × 3 = 729.
Answer: 243, 729.
Example 4: Example 4: Square Numbers
Problem: Write the first 7 square numbers.
Solution:
1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49.
Answer: 1, 4, 9, 16, 25, 36, 49.
Example 5: Example 5: Triangular Numbers
Problem: Find the next two terms: 1, 3, 6, 10, 15, _____, _____.
Solution:
Differences: 3−1=2, 6−3=3, 10−6=4, 15−10=5. Differences increase by 1.
Next: 15 + 6 = 21, 21 + 7 = 28.
Answer: 21, 28.
Example 6: Example 6: Finding a Missing Term
Problem: 8, 15, _____, 29, 36. Find the missing term.
Solution:
Step 1: Difference between known consecutive terms: 36 − 29 = 7.
Step 2: Check: 15 − 8 = 7. Constant difference = 7.
Step 3: Missing term = 15 + 7 = 22.
Answer: The missing term is 22.
Example 7: Example 7: Doubling Sequence
Problem: Ria starts with 1 and doubles each time. Write the first 8 terms.
Solution:
1, 2, 4, 8, 16, 32, 64, 128.
Answer: 1, 2, 4, 8, 16, 32, 64, 128.
Example 8: Example 8: Two-Operation Pattern
Problem: Find the rule and next two terms: 2, 5, 11, 23, 47, _____, _____.
Solution:
5 = 2 × 2 + 1, 11 = 5 × 2 + 1, 23 = 11 × 2 + 1, 47 = 23 × 2 + 1.
Rule: multiply by 2, then add 1.
Next: 47 × 2 + 1 = 95. 95 × 2 + 1 = 191.
Answer: 95, 191.
Example 9: Example 9: Finding the nth Term
Problem: In the sequence 4, 9, 14, 19, 24, ... what is the 10th term?
Solution:
Step 1: Common difference = 5. First term = 4.
Step 2: nth term = first term + (n − 1) × difference
Step 3: 10th term = 4 + (10 − 1) × 5 = 4 + 45 = 49.
Answer: The 10th term is 49.
Example 10: Example 10: Real-Life Sequence
Problem: Arjun saves ₹10 in the first week, ₹20 in the second, ₹30 in the third, and so on. How much does he save in the 8th week?
Solution:
This is an arithmetic sequence: 10, 20, 30, ... with common difference 10.
8th term = 10 + (8 − 1) × 10 = 10 + 70 = 80.
Answer: Arjun saves ₹80 in the 8th week.
Key Points to Remember
- A sequence is an ordered list of numbers following a rule.
- In an arithmetic sequence, the difference between consecutive terms is constant.
- In a geometric sequence, the ratio between consecutive terms is constant.
- To find the rule, look at differences (or ratios) between consecutive terms.
- Square numbers: 1, 4, 9, 16, 25, ... Triangular numbers: 1, 3, 6, 10, 15, ...
- nth term of an arithmetic sequence = first term + (n − 1) × common difference.
- Some sequences use two operations (e.g., multiply then add).
Practice Problems
- Find the next three terms: 7, 14, 21, 28, _____, _____, _____.
- Find the next two terms: 100, 90, 80, 70, _____, _____.
- Find the rule and next two terms: 5, 10, 20, 40, _____, _____.
- What is the 12th term of the sequence 3, 8, 13, 18, 23, ...?
- Find the missing term: 6, 12, _____, 24, 30.
- Write the first 6 triangular numbers.
- Meera adds 15 to a number repeatedly starting from 2. Write the first 5 terms.
- A sequence starts at 1 and each term is tripled. Write the first 6 terms.
Frequently Asked Questions
Q1. What is a number sequence?
A number sequence is an ordered list of numbers that follows a specific rule or pattern. Each number is called a term.
Q2. How do I find the rule of a sequence?
Find the differences between consecutive terms. If the difference is constant, the rule is “add (or subtract) that number.” If the ratio is constant, the rule is “multiply (or divide) by that number.”
Q3. What is an arithmetic sequence?
A sequence where each term is obtained by adding (or subtracting) a fixed number to the previous term. Example: 3, 7, 11, 15 (add 4).
Q4. What is a geometric sequence?
A sequence where each term is obtained by multiplying the previous term by a fixed number. Example: 2, 6, 18, 54 (multiply by 3).
Q5. What are square numbers?
Numbers obtained by multiplying a number by itself: 1, 4, 9, 16, 25, 36, ... (1², 2², 3², etc.).
Q6. What are triangular numbers?
Numbers formed by adding consecutive natural numbers: 1, 1+2=3, 1+2+3=6, 1+2+3+4=10, ... The differences between consecutive terms increase by 1.
Q7. How do I find the nth term of an arithmetic sequence?
Use the formula: nth term = first term + (n − 1) × common difference. For the sequence 5, 8, 11, ..., the 10th term = 5 + 9 × 3 = 32.
Q8. Can a sequence have more than one rule?
Yes. Some sequences use two operations (like multiply by 2 then add 1). Some alternate between adding and multiplying.
Q9. Is this topic in the NCERT Class 5 syllabus?
Yes. Number sequences and patterns are part of the Patterns and Algebra chapter in NCERT/CBSE Class 5 Maths.
