A7LAM CSCA ACADEMYCSCA Academy / Mathematics / Functions
CSCA Mathematics · Lesson 8 of 16Sequences
Suites · المتتاليات
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Sequences are lists of numbers that follow a rule. Two types come up again and again: arithmetic (add the same number) and geometric (multiply by the same number).
Key ideas
- Arithmetic: each term adds d. nth term aₙ = a₁ + (n − 1)d.
- Sum of n arithmetic terms: Sₙ = n(a₁ + aₙ)/2.
- Geometric: each term multiplies by r. nth term aₙ = a₁ · rⁿ⁻¹.
- Sum of n geometric terms (r ≠ 1): Sₙ = a₁(1 − rⁿ)/(1 − r).
Formulas to know
- aₙ = a₁ + (n − 1)d
- aₙ = a₁ · rⁿ⁻¹
- 1 + 2 + … + n = n(n + 1)/2
Worked example
Find the 10th term of 3, 7, 11, 15, …
- It is arithmetic with a₁ = 3 and d = 4.
- a₁₀ = 3 + 9 · 4.
Answer: a₁₀ = 39
Practice questions
1. Next term of 2, 6, 18, 54, …?
108 · 162 ✓ · 72 · 216
Geometric with r = 3: 54 × 3 = 162.
2. 1 + 2 + 3 + … + 100 = ?
5,000 · 5,050 ✓ · 10,100 · 4,950
n(n + 1)/2 = 100 × 101 / 2 = 5,050.
3. An arithmetic sequence has a₁ = 5 and a₄ = 20. Find d.
3 · 5 ✓ · 15 · 4
a₄ = a₁ + 3d, so 20 = 5 + 3d and d = 5.