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

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Grade 12 · Patterns & Sequences

Series, Sigma Notation & Convergence (Grade 12)

MEMORANDUM

Name:

Class:

Date:

Mark

/ 16

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.
(2)
2(10) + 4 = 24✓✓ (2)
2
Determine the sum of the first 19 terms of Tₙ = 2n + 4.
(2)
a = 6, d = 2. S₁₉ = 192[12 + 36] = 456✓✓ (2)
3
Evaluate ∑_n=1²⁰ (3n − 1)
(2)
a = 2, d = 3, n = 20. S = 10[4 + 57] = 610✓✓ (2)
4
How many terms are there in ∑_n=3¹⁰ Tₙ?
(2)
10 − 3 + 1 = 8 terms✓✓ (2)
5
Write 2 + 5 + 8 + … + 59 in sigma notation.
(2)
Tₙ = 3n − 1 and 3n − 1 = 59 gives n = 20, so ∑_n=1²⁰ (3n − 1)✓✓ (2)
6
Determine S_∞ for 8 + 4 + 2 + 1 + …
(2)
r = 12, so S_∞ = 81 − 12 = 16✓✓ (2)
7
Determine the values of x for which ∑ 3(2x)ⁿ⁻¹ converges.
(2)
−1 < 2x < 1, so −12 < x < 12✓✓ (2)
8
Calculate S₁₀ for the geometric series 2 + 6 + 18 + …
(2)
a = 2, r = 3. S₁₀ = 2(3¹⁰ − 1)2 = 59 048✓✓ (2)

TOTAL: 16 marks