Series, Sigma Notation & Convergence (Grade 12) — Worksheet & Memo

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Grade 12 · Number Patterns, Sequences and Series

Series, Sigma Notation & Convergence (Grade 12)

MEMORANDUM

Name:

Class:

Date:

Mark

/ 29

Answer all questions. Show all your working — marks are awarded for method as well as the final answer. Teacher copy: accept any correct equivalent method.

1
The general term of an arithmetic sequence is Tₙ = 2n + 4. Determine T₁₀.
(1)
2(10) + 4 = 24✓ (1)
2
Determine the sum of the first 19 terms of Tₙ = 2n + 4.
(3)
a = 6, d = 2. S₁₉ = 19/2[12 + 36] = 456✓✓✓ (3)
3
Evaluate ∑_n=1²⁰ (3n − 1)
(3)
a = 2, d = 3, n = 20. S = 10[4 + 57] = 610✓✓✓ (3)
4
How many terms are there in ∑_n=3¹⁰ Tₙ?
(1)
10 − 3 + 1 = 8 terms✓ (1)
5
Write 2 + 5 + 8 + … + 59 in sigma notation.
(3)
Tₙ = 3n − 1 and 3n − 1 = 59 gives n = 20, so ∑_n=1²⁰ (3n − 1)✓✓✓ (3)
6
Determine S_∞ for 8 + 4 + 2 + 1 + …
(2)
r = 1/2, so S_∞ = 8/(1 − 1/2) = 16✓✓ (2)
7
Determine the values of x for which ∑ 3(2x)^{n − 1} converges.
(3)
−1 < 2x < 1, so −1/2 < x < 1/2✓✓✓ (3)
8
Calculate S₁₀ for the geometric series 2 + 6 + 18 + …
(3)
a = 2, r = 3. S₁₀ = (2(3¹⁰ − 1))/2 = 59 048✓✓✓ (3)
9
Derive the formula Sₙ = n/2 (a + l) for an arithmetic series.
(4)
Write Sₙ forwards and backwards ✓; adding gives 2Sₙ = n(a + l) because each pair sums to a + l ✓✓; ∴ Sₙ = n/2 (a + l) ✓(4)
10
Derive the formula for the sum of the first n terms of a geometric series.
(4)
Sₙ = a + ar + … + arⁿ⁻¹; multiply by r ✓; subtract to get rSₙ − Sₙ = arⁿ − a ✓; factorise: Sₙ(r − 1) = a(rⁿ − 1) ✓; Sₙ = a(rⁿ − 1)/(r − 1), r ≠ 1 ✓(4)
11
Explain why the sum to infinity only exists when −1 < r < 1.
(2)
Only then does rⁿ approach 0 as n grows ✓, so the terms shrink and the total settles on a finite value ✓(2)

TOTAL: 29 marks