ZIMSEC A Level · Pure Mathematics · Form 5
Arithmetic progressions (AP)
uₙ = a + (n − 1)d; Sₙ = (n/2)[2a + (n − 1)d] = (n/2)(a + l).
Geometric progressions (GP)
uₙ = arⁿ⁻¹; Sₙ = a(1 − rⁿ)/(1 − r); sum to infinity S∞ = a/(1 − r) when |r| < 1.
Worked example: A GP has first term 24 and sum to infinity 36. Find the common ratio.
36 = 24/(1 − r) → 1 − r = 2/3 → r = 1/3
Sigma notation
Σ from r = 1 to n of r = n(n + 1)/2.
Binomial expansion
- Positive integer n: (a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ, where ⁿCᵣ = n!/(r!(n − r)!).
- Any rational n: (1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + n(n − 1)(n − 2)x³/3! + …, valid for |x| < 1.
Worked example: Expand (1 − 2x)^(1/2) up to the term in x², and state the range of validity.
1 + ½(−2x) + [½ × (−½)/2](−2x)² = 1 − x − ½x²
Valid for |2x| < 1, i.e. |x| < ½
Worked example: Find the term in x³ in (2 + x)⁵.
⁵C₃ × 2² × x³ = 10 × 4x³ = 40x³
Exam tip: For non-integer powers, write the expression as k(1 + …)ⁿ first and always state the validity.
Check yourself
- An AP has a = 5 and d = 3. Find S₂₀.
- Find the sum to infinity of 8 + 4 + 2 + …
- Find the first three terms of (1 + x)⁻² in ascending powers of x.
Show answers
- 10(10 + 57) = 670
- 8/(1 − ½) = 16
- 1 − 2x + 3x²
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