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

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Grade 12 · Patterns & Sequences

Series, Sigma Notation & Convergence (Grade 12)

EXAM-STYLE CLASS TEST

Marks

Duration

50 minutes

Questions

Name:

Class:

Date:

Mark

/ 32

Instructions and Information

Answer ALL the questions in this question paper.
Answer QUESTION 1 by circling the letter (A–D) in the answer grid at the end of that section.
Show ALL calculations clearly.
Show all units where applicable.
Number the answers correctly according to the numbering system used in this question paper.
A non-programmable calculator may be used, unless stated otherwise.
Write neatly and legibly.

Question 1

[10 MARKS]

Four options are given as possible answers to the following questions. Choose the correct answer and circle the letter (A–D) in the grid at the end of this section. If you want to change your choice, put a cross through the wrong letter and circle your new choice.

1.1
The value of the sum of 2k, for k running from 1 to 5, is …
(1)
A)30
B)10
C)15
D)25
1.2
In sigma notation, the number written BELOW the sigma is …
(1)
A)the first value of the counter
B)the number of terms
C)the last value of the counter
D)the common difference
1.3
A sum runs with its counter from k = 3 to k = 20. The number of terms is …
(1)
A)18
B)17
C)20
D)23
1.4
A geometric series CONVERGES when …
(1)
A)−1 < r < 1
B)r > 1
C)r ≥ 1
D)r ≤ −1
1.5
The sum to infinity of 12 + 6 + 3 + … is …
(1)
A)24
B)21
C)12
D)48
1.6
For which values of x does 1 + x + x² + x³ + … converge?
(1)
A)−1 < x < 1
B)x > 1
C)x is not 0
D)all real x
1.7
The value of the sum of 3k, for k running from 1 to 10, is …
(1)
A)165
B)55
C)30
D)150
1.8
The value of the sum of the CONSTANT 4, for k running from 1 to 7, is …
(1)
A)28
B)4
C)7
D)11
1.9
A SERIES is the … of a sequence.
(1)
A)sum
B)list
C)ratio
D)difference
1.10
If S_∞ = 40 and a = 10, then r = …
(1)
A)0,75
B)0,25
C)4
D)30

Answer grid — circle your answers for Question 1

1.1	A	B	C	D
1.2	A	B	C	D
1.3	A	B	C	D
1.4	A	B	C	D
1.5	A	B	C	D
1.6	A	B	C	D
1.7	A	B	C	D
1.8	A	B	C	D
1.9	A	B	C	D
1.10	A	B	C	D

Question 2

[9 MARKS]

Consider the series 3 + 9 + 15 + … + 273.

2.1
Determine how many terms there are in the series.
(3)
2.2
Determine the sum of the series.
(3)
2.3
Write the series in sigma notation.
(3)

Question 3

[13 MARKS]

Answer the questions below on geometric series.

3.1
Determine the value of n if Σ_k=3ⁿ 8(2)^k−1 = 131 040.
(5)
3.2
Gold is extracted from an old mine heap. Processing 1 000 tons of gravel yields 30 kg of gold the first time, 24 kg the second, 19,2 kg the third and so on. Determine the MAXIMUM amount of gold that can be recovered.
(4)
3.3
Determine the values of x for which the series Σ3(2x)ⁿ⁻¹ converges.
(4)

TOTAL: 32 marks

This question paper consists of 3 questions.