Arithmetic and geometric sequences
The two reference sequences: definition by recurrence, common difference or ratio, general term, monotonicity and sum of the first terms, then spotting them in a situation (simple or compound interest, doubling, depreciation).
- 26 cards
- 6 topics
- Free
The outline, topic by topic
Each topic is weighted the way the exam weighs it.
- Concepts
- Questions
- Sort and link
- The numbers
- Scenarios
- Understand
A few cards from the deck
Arithmetic sequence: what is it?
Show the answer
A sequence in which you get from each term to the next by always adding the same number.
The general term is uₙ = u₀ + n × r: the points (n, uₙ) lie on a straight line.
What does this program print?
```python
u = 3
for _ in range(4):
u = 2 * u
print(u)`48249611
Show the answer
48The loop multiplies by 2 four times: 6, 12, 24, 48. The term obtained is u₄ = 3 × 2⁴ = 48. Adding 2 four times would give 11, and only three rounds 24.
For an arithmetic sequence with common difference r, the general term is uₙ = …; for a geometric sequence with ratio q, uₙ = ….
Show the answer
For an arithmetic sequence with common difference r, the general term is uₙ = u₀ + n × r; for a geometric sequence with ratio q, uₙ = u₀ × qⁿ.
You add r n times in one case, multiply by q n times in the other. More generally, uₙ = uₚ + (n − p) × r, or uₙ = uₚ × qⁿ⁻ᵖ.
Geometric sequence: what is it?
Show the answer
A sequence in which you get from each term to the next by always multiplying by the same number.
The general term is uₙ = u₀ × qⁿ: the terms change by a constant percentage, hence exponential growth or decay.