Learn how constant steps create predictable patterns, model growth, and unlock the rest of the sequences and series unit.
Think about a stack of bricks, a running program that adds 200 m every week, or wages that increase by R150 per month. Every scenario adds a constant amount, so you can predict the future by counting equal steps. That constant-step idea is the backbone of arithmetic sequences.
Rise = 18 cm per step → after 7 steps you climbed 6 × 18 cm above the first step.
That is exactly the formula $$a_7 = a_1 + 6d$$ in action.
Common pitfall
Students often count the number of terms incorrectly. Remember: $$n - 1$$ counts the gaps between terms, not the terms themselves.
Start by comparing successive terms. If the difference is the same every time, you are dealing with an arithmetic sequence.
Checklist
Formula
a_n = 3 + (n - 1)(2)
Highlighted Term
a5 = 11
Change
Increasing by 2 each step
What to notice
Each term equals the starting value plus a certain number of jumps. The number of jumps is always one less than the term number because the first term requires zero jumps.
nth-term calculation
Sum of first n terms
S_n = n(a_1 + a_n) / 2
S_6 = 6(4 + 19) / 2 = 69.0
First 6 terms
Interpretation
The expression a_n = a_1 + (n - 1)d simply counts how many steps of size d you have taken from the starting point.
If you track totals (useful in finance problems), the sum formula averages the first and last terms before multiplying by the number of terms.
Saving plans, production targets, or revision schedules that add a fixed amount every period are all arithmetic sequences. Build and test one here.
Formula snapshot
Deposit_k = 80 + (k - 1)(15)
Focus week amount
R125.00
Total saved
R705.00
| Week | Amount (R) | How far from start? |
|---|---|---|
| Week 1 | 80.00 | starting value |
| Week 2 | 95.00 | 1 · d = 1 · 15 |
| Week 3 | 110.00 | 2 · d = 2 · 15 |
| Week 4 | 125.00 | 3 · d = 3 · 15 |
| Week 5 | 140.00 | 4 · d = 4 · 15 |
| Week 6 | 155.00 | 5 · d = 5 · 15 |
Link to real problems
Any situation with constant increases (training distances, wages, payments, production targets) can be modelled exactly like this. The sliders let you test "what if" questions before committing to a plan.
Work through these problems. Compare your answers with the formulas above before revealing the full solutions.
Test how well you can interpret and manipulate arithmetic sequences without help.
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