IB MYP 4-5 Maths- Number sequences - Study Notes - New Syllabus
IB MYP 4-5 Maths- Number sequences – Study Notes
Standard
- Number sequences
IB MYP 4-5 Maths- Number sequences – Study Notes – All topics
Number Sequences (Prediction & Description)
Sequence
A sequence is an ordered list of numbers that follow a particular rule or pattern. Each number in the sequence is called a term, and the position of the term is called its order.
For example: \( 2,\ 4,\ 6,\ 8,\ 10,\ldots \) is a sequence where each number increases by 2.
Order of the Sequence
The order tells us the position of a number in the sequence.
- The first term is usually denoted by \( T_1 \)
- The second term is \( T_2 \), and so on
- The nth term is \( T_n \), which represents any general term in the sequence
Types of Order:
- Ascending order: Each term is larger than the previous one (increasing)
Example: \( 3,\ 6,\ 9,\ 12,\ldots \) - Descending order: Each term is smaller than the previous one (decreasing)
Example: \( 100,\ 90,\ 80,\ 70,\ldots \)
Finite and Infinite Sequences
- Finite sequence: A sequence that has a fixed number of terms.
Example: \( 2,\ 4,\ 6,\ 8 \) (only 4 terms) - Infinite sequence: A sequence that continues forever with no end.
Example: \( 1,\ 2,\ 3,\ 4,\ 5,\ldots \)
Types of Sequences in Math
- ① Arithmetic Sequence
- ② Quadratic Sequence
- ③ Geometric Sequence
- ④ Fibonacci Sequence
- ⑤ Harmonic Sequence
- ⑥ Triangular Number Sequence
- ⑦ Square Number Sequence
- ⑧ Cube Number Sequence
In MYP Mathematics, students explore different types of sequences, how to identify their patterns, write rules, and predict future terms using general formulas. Understanding the type of sequence helps in writing the rule and solving problems efficiently.
1. Arithmetic Sequence
An arithmetic sequence has a constant difference between terms.
General rule: \( T_n = a + (n – 1)d \)
- \( a \): first term
- \( d \): common difference
Example: Find the 12th term of the sequence: 5, 8, 11, 14, …
▶️ Answer/Explanation
\( a = 5,\ d = 3 \)
\( T_{12} = 5 + (12 – 1) \cdot 3 = 5 + 33 = \boxed{38} \)
Example: Write the nth term of the sequence: -4, -1, 2, 5, …
▶️ Answer/Explanation
\( d = 3,\ a = -4 \)
\( T_n = -4 + (n – 1) \cdot 3 = \boxed{3n – 7} \)
2. Quadratic Sequence
In a quadratic sequence, the difference between terms is not constant, but the second difference is constant.
The nth term is of the form: \( T_n = an^2 + bn + c \)
Example : Find the nth term of: 1, 4, 9, 16, 25, …
▶️ Answer/Explanation
This is \( n^2 \) → \( T_n = \boxed{n^2} \)
Example : Find the next term: 3, 8, 15, 24, 35, …
▶️ Answer/Explanation
Second differences are constant → Quadratic
Pattern: \( T_n = n^2 + 2 \)
Next term: \( T_6 = 6^2 + 2 = \boxed{38} \)
3. Geometric Sequence
Each term is multiplied by a fixed number (common ratio).
Rule: \( T_n = a \cdot r^{n-1} \)
Example : Find the 5th term of the sequence: 2, 6, 18, 54, …
▶️ Answer/Explanation
\( a = 2,\ r = 3 \)
\( T_5 = 2 \cdot 3^4 = \boxed{162} \)
Example : Write the nth term of: 5, 10, 20, 40, …
▶️ Answer/Explanation
\( a = 5,\ r = 2 \)
\( T_n = 5 \cdot 2^{n-1} \)
4. Fibonacci Sequence
Each term is the sum of the two preceding terms.
Rule: \( T_n = T_{n-1} + T_{n-2} \)
Example : Continue the sequence: 1, 1, 2, 3, 5, 8, …
▶️ Answer/Explanation
Next term = 5 + 8 = \boxed{13}
Example : If the first two terms are 2 and 5, find the next 3 terms.
▶️ Answer/Explanation
Next: \( 2+5=7,\ 5+7=12,\ 7+12=19 \)
Sequence: 2, 5, 7, 12, 19
5. Harmonic Sequence
The reciprocals of the terms form an arithmetic sequence.
If \( \frac{1}{T_n} \) is arithmetic, then it’s harmonic.
Example : Find the next term: 1, \( \frac{1}{2} \), \( \frac{1}{3} \), \( \frac{1}{4} \), …
▶️ Answer/Explanation
Next term is \( \frac{1}{5} \)
Example : Find the nth term of the harmonic sequence: \( \frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \ldots \)
▶️ Answer/Explanation
The underlying arithmetic: 2, 4, 6, …
So \( T_n = \frac{1}{2n} \)
6. Triangular Number Sequence
These are the sums of the natural numbers.
Rule: \( T_n = \frac{n(n + 1)}{2} \)
Example : Find the 6th triangular number.
▶️ Answer/Explanation
\( T_6 = \frac{6 \cdot 7}{2} = \boxed{21} \)
Example : Check if 36 is a triangular number.
▶️ Answer/Explanation
Try values of \( n \): \( T_8 = \frac{8 \cdot 9}{2} = 36 \)
Yes, \( \boxed{n = 8} \)
7. Square Number Sequence
Each term is a perfect square.
Rule: \( T_n = n^2 \)
Example : Find the 9th square number.
▶️ Answer/Explanation
\( T_9 = 9^2 = \boxed{81} \)
Example : Find the nth term of: 1, 4, 9, 16, …
▶️ Answer/Explanation
\( T_n = n^2 \)
8. Cube Number Sequence
Each term is a perfect cube.
Rule: \( T_n = n^3 \)
Example : Find the 5th cube number.
▶️ Answer/Explanation
\( T_5 = 5^3 = \boxed{125} \)
Example : Predict the next term in the sequence: 1, 8, 27, 64, …
▶️ Answer/Explanation
Next term = \( 5^3 = \boxed{125} \)
Example :
The first five terms of a sequence are: 3, 7, 11, 15, 19
Part a) Describe the rule for this sequence.
Part b) Find the 10th term.
▶️ Answer/Explanation
Part a: Identify the pattern
Each term increases by 4 → This is an arithmetic sequence.
Rule: Add 4 each time or \( T_n = 3 + (n – 1) \cdot 4 \)
Part b: Use the nth term formula
\( T_{10} = 3 + (10 – 1) \cdot 4 = 3 + 36 = 39 \)
Example :
The sequence is: 2, 6, 18, 54, …
Part a) What type of sequence is this?
Part b) What is the 6th term?
▶️ Answer/Explanation
Part a: Identify the pattern
Each term is multiplied by 3 → Geometric sequence
Common ratio \( r = 3 \), first term \( a = 2 \)
Part b: Use formula \( T_n = a \cdot r^{n-1} \)
\( T_6 = 2 \cdot 3^5 = 2 \cdot 243 = 486 \)
Example :
The sequence of square numbers is given as: 1, 4, 9, 16, 25, …
Part a) Find the nth term rule.
Part b) Predict the 12th term.
▶️ Answer/Explanation
Part a:
This sequence follows: \( 1^2, 2^2, 3^2, … \)
So the rule is \( T_n = n^2 \)
Part b:
\( T_{12} = 12^2 = 144 \)
Example :
A pattern is made using matchsticks to form squares in a row:
1 square uses 4 sticks
2 squares use 7 sticks
3 squares use 10 sticks
Part a) Write a rule for the number of sticks (S) for \( n \) squares.
Part b) Predict how many sticks are needed for 20 squares.
▶️ Answer/Explanation
Part a: Find the pattern
Each new square adds 3 sticks (shares 1 side with previous)
So rule is: \( S = 3n + 1 \)
Part b:
For 20 squares: \( S = 3(20) + 1 = 60 + 1 = 61 \)