Arithmetic and Geometric Sequences

1. 1 Display several sequences. Students classify as arithmetic (constant difference), geometric (constant ratio), or neither: 3,7,11,15 (arithmetic, d=4); 2,6,18,54 (geometric, r=3); 1,4,9,16 (neither — square numbers); 100,10,1,0.1 (geometric, r=0.1).
2. 2 nth term: uₙ = a + (n−1)d, where a = first term, d = common difference. Sum of n terms: Sₙ = n/2(2a+(n−1)d) or Sₙ = n/2(a+l) where l = last term. Students find the 20th term and sum of first 20 terms for 5, 8, 11, 14...
3. 3 nth term: uₙ = arⁿ⁻¹. Sum: Sₙ = a(1−rⁿ)/(1−r) for r≠1. For 3, 6, 12, 24... find u₈ and S₈. Students apply to another geometric sequence with r = 0.5.
4. 4 For |r| < 1, the sum converges: S∞ = a/(1−r). For 4, 2, 1, 0.5... a=4, r=0.5. S∞ = 4/(1−0.5) = 8. 'Intuitively, the terms get so small they eventually contribute nothing.' Students find S∞ for three convergent series.
5. 5 Compound interest: deposit £1000 at 4% per year. Year 1: £1040, Year 2: £1081.60... Geometric sequence with a=1000, r=1.04. How much after 10 years? (u₁₀ = 1000 × 1.04⁹ ≈ £1423.) How long until it doubles? (Solve 2000 = 1000 × 1.04ⁿ⁻¹.)
6. 6 An arithmetic sequence has 3rd term 11 and 7th term 23. Find a and d. Set up simultaneous equations from uₙ formula. Solve. A geometric sequence has u₂ = 6 and u₅ = 48. Find a and r. Students solve both.
7. 7 Prove that the sum of the first n odd numbers is n². S = 1+3+5+...+(2n−1). Arithmetic series with a=1, d=2: Sₙ = n/2(2+2(n−1)) = n/2(2n) = n². Elegant proof — discuss with students.

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