| Topic | Description |
|---|---|
| Arithmetic Sequences | Sequences where the difference between consecutive terms is constant. |
| nth Term Formula | Formula used to calculate any term directly. |
| Arithmetic Series | The sum of the first n terms of an arithmetic sequence. |
| Sigma Notation | Compact notation representing repeated addition. |
📌 Understanding Arithmetic Sequences
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference results in a linear pattern of growth or decay.
a, a + d, a + 2d, a + 3d, …
- a represents the first term.
- d represents the common difference.
- Each term increases or decreases by the same fixed amount.
- The sequence can be increasing (d > 0) or decreasing (d < 0).
🌍 Real-World Connection
- Annual salary increments that rise by a fixed amount.
- Linear depreciation of vehicles and machinery.
- Distance covered at constant speed.
📌 The nth Term Formula
The nth term formula allows the calculation of any term without listing all previous terms, making it essential for large values of n.
un = a + (n − 1)d
- Widely used in IB Paper 1 algebraic sequence problems.
- Allows formation of simultaneous equations.
- Helps determine whether a number belongs to a sequence.
🧠 Examiner Tip
- Always substitute values clearly.
- Two terms → two equations.
- State answers in exact form where possible.
📌 Arithmetic Series
An arithmetic series is the sum of the first n terms of an arithmetic sequence.
Sn = n/2 (2a + (n − 1)d)
- Used when total accumulation is required.
- Often tested in structured IB questions.
- Can also be expressed using the last term.
📝 Paper 1 Strategy
- Identify a, d, and n before substituting.
- Show full substitution for method marks.
- Simplify only at the final step.
📌 Practice Questions
Multiple Choice Questions (MCQs)
MCQ 1. Find the common difference of the arithmetic sequence:
7, 12, 17, 22
A. 3
B. 4
C. 5
D. 7
Answer: C
Explanation: Each term increases by 5, so the common difference d = 5.
MCQ 2. What is the 8th term of the arithmetic sequence:
3, 7, 11, …
A. 27
B. 29
C. 31
D. 35
Answer: C
Explanation:
a = 3, d = 4
u8 = 3 + (8 − 1)×4 = 3 + 28 = 31.
MCQ 3. Which of the following sequences is arithmetic?
A. 2, 4, 8, 16
B. 1, 4, 9, 16
C. 5, 10, 15, 20
D. 3, 6, 12, 24
Answer: C
Explanation: Sequence C has a constant difference of 5.
The others involve multiplication or changing differences.
MCQ 4. A sequence has a negative common difference. What does this imply?
A. The sequence is geometric
B. The sequence is decreasing
C. The sequence is increasing
D. The sequence is non-linear
Answer: B
Explanation: A negative value of d means each term is smaller than the previous term, so the sequence decreases.
SAQ 1. Find the 15th term of the sequence 2, 6, 10…
Worked Answer:
a = 2, d = 4
u15 = 2 + 14×4 = 58
SAQ 2. Determine whether 101 is a term of the sequence 5, 9, 13…
Worked Answer:
101 = 5 + (n − 1)4
96 = 4(n − 1) → n = 25
Therefore, 101 is a term.
Long Answer / Explainer Questions
Q7.
The first term of an arithmetic sequence is 6 and the common difference is 4.
(a) Derive an expression for the nth term of the sequence.
(b) Determine whether 406 is a term of the sequence.
(c) Find the number of terms in the sequence that are less than 500.
(b) Determine whether 406 is a term of the sequence.
(c) Find the number of terms in the sequence that are less than 500.
Full Worked Solution:
(a) nth term
For an arithmetic sequence, the nth term is given by:
un = a + (n − 1)d
Substituting a = 6 and d = 4:
un = 6 + 4(n − 1)
un = 4n + 2
(b) Checking whether 406 is a term
To determine whether 406 is a term, equate the nth term to 406:
4n + 2 = 406
4n = 404 → n = 101
Since n is a positive integer, 406 is a term of the sequence.
(c) Number of terms less than 500
We solve the inequality:
4n + 2 < 500
4n < 498 → n < 124.5
The greatest integer value of n is 124.
Conclusion:
- 406 is the 101st term.
- There are 124 terms less than 500.
Q8.
An arithmetic sequence has a first term of 12 and a common difference of −3.
(a) Explain why the sequence is decreasing.
(b) Find the sum of the first 30 terms.
(c) Interpret the meaning of this result in a real-world context.
(b) Find the sum of the first 30 terms.
(c) Interpret the meaning of this result in a real-world context.
Full Worked Solution:
(a) Nature of the sequence
The common difference is −3, which means that each term is obtained by subtracting 3 from the previous term.
Since each new term is smaller than the one before it, the sequence is strictly decreasing.
(b) Sum of the first 30 terms
The formula for the sum of the first n terms of an arithmetic sequence is:
Sn = n/2 (2a + (n − 1)d)
Substituting a = 12, d = −3, and n = 30:
S30 = 30/2 [2(12) + 29(−3)]
S30 = 15 (24 − 87)
S30 = 15 (−63) = −945
(c) Interpretation
A negative sum indicates that, over time, the total change is a net decrease.
In a real-world context, this could represent:
- A steadily decreasing bank balance with fixed withdrawals.
- Consistent loss of altitude or energy over time.
- Depreciation where losses outweigh initial value.
Thus, arithmetic series not only compute totals but also provide insight into long-term trends.
📐 IA Spotlight
Use arithmetic sequences to model real-world linear trends, then analyse where the linear model becomes unrealistic.