Arithmetic & Geometric Sequences: Full Guide — Worksheet & Memo

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Grade 12 · Number Patterns, Sequences and Series

Arithmetic & Geometric Sequences: Full Guide

MARKING GUIDELINE

Marks

Duration

50 minutes

Questions

Name:

Class:

Date:

Mark

/ 33

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
T_n = a + (n − 1)d is the general term of …
(1)
A)a quadratic pattern
B)a geometric sequence
C)an arithmetic sequence✓
D)a convergent series
Answer: C — A common DIFFERENCE d is what makes a sequence arithmetic.
- A — a quadratic pattern has an n² term
- B — a geometric sequence uses T_n = ar^{n − 1}
- D — a series is a SUM, not a sequence
1.2
The 20th term of 7 ; 11 ; 15 ; … is …
(1)
A)87
B)83✓
C)80
D)76
Answer: B — T₂₀ = 7 + 19(4) = 83.
- A — used 20d instead of 19d
- C — used 20d and then dropped the first term
- D — used 19d without adding the first term
1.3
The common ratio of 3 ; 6 ; 12 ; … is …
(1)
A)3
B)6
C)1/2
D)2✓
Answer: D — Each term is DIVIDED by the one before: 6 ÷ 3 = 2.
- A — that is the first term
- B — SUBTRACTED consecutive terms instead of dividing them
- C — inverted the ratio
1.4
The 8th term of 3 ; 6 ; 12 ; … is …
(1)
A)768
B)192
C)384✓
D)24
Answer: C — T₈ = 3(2)⁷ = 384.
- A — used r⁸ instead of r⁷
- B — used r⁶ instead of r⁷
- D — worked out 3 × 8 instead of 3 × 2⁷
1.5
The sum of the first 20 terms of 7 ; 11 ; 15 ; … is …
(1)
A)1 800
B)830
C)90
D)900✓
Answer: D — S_n = n/2[2a + (n − 1)d] = 10(14 + 76) = 900.
- A — left out the factor of a half
- B — used n/2 × T₂₀, leaving out the first term
- C — left off the factor of 10 altogether
1.6
In an arithmetic sequence T₅ = 17 and T₉ = 33. Then d = …
(1)
A)8
B)4✓
C)16
D)2
Answer: B — Four steps separate T₅ from T₉, so 4d = 16 and d = 4.
- A — divided by 2 instead of by 4
- C — that is 4d, before dividing by the four steps
- D — divided by 8 instead of by 4
1.7
A geometric SERIES converges when …
(1)
A)−1 < r < 1✓
B)r > 1
C)r < 0
D)r = 1
Answer: A — Only then do the terms shrink fast enough for the sum to settle on a value.
- B — the terms grow without limit, so the sum diverges
- C — r = −2 is negative, but that series still diverges
- D — every term is then the same, so the sum grows without limit
1.8
The sum to infinity of 8 + 4 + 2 + … is …
(1)
A)8
B)14
C)16✓
D)32
Answer: C — S_∞ = a ÷ (1 − r) = 8 ÷ 0,5 = 16.
- A — that is the first term alone
- B — added the first three terms and stopped
- D — used 1 − r as 0,25 instead of 0,5
1.9
In a geometric sequence T₃ = 20 and T₅ = 80. The POSITIVE value of r is …
(1)
A)4
B)2✓
C)60
D)√60
Answer: B — T₅ = T₃r², so r² = 80 ÷ 20 = 4 and r = 2.
- A — that is r², before taking the square root
- C — SUBTRACTED the two terms instead of dividing
- D — square-rooted the difference instead of the ratio
1.10
The sum of the first n terms of a GEOMETRIC series (r − 1 is not 0) is …
(1)
A)(a(rⁿ − 1))/(r − 1)✓
B)a/(1 − r)
C)(a(rⁿ + 1))/(r − 1)
D)n/2[2a + (n − 1)d]
Answer: A — That is the geometric sum formula from the information sheet.
- B — that is the sum to INFINITY, not the sum of n terms
- C — the numerator carries a MINUS, not a plus
- D — that is the ARITHMETIC sum formula

Answer grid — marking guideline

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

[11 MARKS]

In an ARITHMETIC sequence the third term is 11 and the seventh term is 27.

2.1
Determine the common difference d.
(3)
T₇ − T₃ = 4d (1)
27 − 11 = 4d (1)
d = 4 (1)
2.2
Determine the first term a.
(2)
T₃ = a + 2d: 11 = a + 8 (1)
a = 3 (1)
2.3
Calculate the sum of the first 20 terms.
(3)
S₂₀ = 20/2[2(3) + 19(4)] (2) = 820 (1)
2.4
Determine which term of the sequence is equal to 99.
(3)
3 + (n − 1)4 = 99 (1)
4n − 1 = 99 (1)
n = 25 (1)

Question 3

[12 MARKS]

In a GEOMETRIC sequence the second term is 6 and the fifth term is 48.

3.1
Determine the constant ratio r.
(4)
T₅/T₂ = r³ (1)
48/6 = 8 (1)
r³ = 8 (1)
r = 2 (1)
3.2
Determine the first term.
(2)
T₂ = ar: 6 = 2a (1)
a = 3 (1)
3.3
Calculate the sum of the first 8 terms.
(3)
S₈ = (3(2⁸ − 1))/(2 − 1) (2) = 765 (1)
3.4
Explain why this series does NOT have a sum to infinity, and give an example of a geometric series that does.
(3)
S_∞ exists only when |r| < 1 (1), and here r = 2 (1)
For example 1/2 + 1/4 + 1/8 + …, where r = 1/2 and S_∞ = 1 (1)

TOTAL: 33 marks

This question paper consists of 3 questions.