| Topic | Description |
|---|---|
| Arithmetic Sequences | Sequences with a constant difference between successive terms. |
| nth Term Formula | A rule to find any term of an arithmetic sequence without listing previous terms. |
| Summation Formula (Series) | Used to calculate the sum of the first n terms of an arithmetic sequence. |
| Sigma Notation | Compact mathematical notation used to represent sums. |
| Applications | Growth models, savings, simple interest, salary increases, predictable patterns. |
1. Understanding Arithmetic Sequences
An arithmetic sequence is defined by a constant difference between consecutive terms.
If the first term is a and the common difference is d, the sequence looks like:
a, a + d, a + 2d, a + 3d, ā¦
This constant difference makes arithmetic sequences one of the simplest models of steady change.
They are widely used in finance, population modelling, salary growth, and linear approximations in science.
Many incomes increase by the same fixed amount every year rather than by percentage (e.g., +ā¹10,000 per year).
This makes real salaries closer to arithmetic sequences, not geometric ones.
Why do humans find patterns so compelling?
Arithmetic sequences reflect our natural desire to see regularity ā but in reality, almost no situation behaves perfectly linearly.
2. The nth Term Formula
The nth term tells us the value of any term in the sequence without manually generating earlier ones.
The formula is:
uā = a + (n ā 1)d
Example:
Sequence: 7, 12, 17, 22, ā¦
Here a = 7 and d = 5.
The 20th term is uāā = 7 + 19Ć5 = 102.
If a question gives two specific terms (like uā and uāā), always form two equations and solve simultaneously for a and d.
Indian mathematician Aryabhata described arithmetic progressions over 1500 years ago.
Ancient chess legends (e.g., Sissa ibn Dahir) also use structured numeric patterns that relate to sequence theory.
3. Arithmetic Series: Sum of the First n Terms
An arithmetic series is the sum of terms from an arithmetic sequence.
There are two equivalent formulas:
Sā = n/2 (2a + (n ā 1)d)
Sā = n/2 (a + uā)
The first version is useful when you know a and d.
The second is useful when you know the last term uā.
Example:
Find the sum of the first 50 terms of 4, 9, 14, 19ā¦
a = 4, d = 5
Sā
ā = 50/2 Ć (2Ć4 + 49Ć5) = 25 Ć 253 = 6325.
If you see sigma notation Ī£(3n + 7), rewrite it as an arithmetic sequence with a = 10 and d = 3, then use the series formula.
4. Sigma Notation (Ī£)
Sigma notation expresses long sums compactly.
For example:
Ī£ (2n + 1) from n = 1 to 5 represents:
(2Ć1 + 1) + (2Ć2 + 1) + (2Ć3 + 1) + (2Ć4 + 1) + (2Ć5 + 1) = 35.
Most arithmetic-sum questions require identifying:
- the first term a,
- the common difference d,
- the number of terms n.
Summing the number of tiles in a stepped architectural structure or calculating total savings with fixed yearly deposits
both naturally use arithmetic series and sigma notation.
5. Applications in Modelling & Prediction
Arithmetic sequences help model situations of steady linear growth.
Examples include:
- simple interest accumulation
- salary increments by fixed amounts
- transport schedules
- production planning
However, many real-life patterns aren’t perfectly linear ā
so you may need to approximate a common difference or recognise when the arithmetic model breaks down.
If your IA investigates growth (finance, sports, population), try fitting both arithmetic and geometric models.
Comparing them can strengthen your Mathematical Communication marks.
Is identifying a pattern the same as explaining it?
Arithmetic sequences describe behaviour ā but do they cause it?
Consider also Fibonacci patterns and the golden ratio as contrasting nonlinear sequences.