Quadratic Number Patterns (Grade 11 & 12) — Worksheet & Memo

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Grade 12 · Patterns & Sequences

Quadratic Number Patterns (Grade 11 & 12)

MARKING GUIDELINE

Marks

Duration

55 minutes

Questions

Name:

Class:

Date:

Mark

/ 34

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
In a QUADRATIC number pattern, the second difference is …
(1)
A)constant✓
B)zero
C)increasing
D)equal to the first difference
Answer: A — A constant second difference is exactly what makes a pattern quadratic.
- B — a zero second difference makes the pattern LINEAR
- C — an increasing second difference means a cubic pattern or higher
- D — the first differences change; the second ones do not
1.2
For the pattern 204 ; 176 ; 150 ; 126 ; …, the SECOND difference is …
(1)
A)2✓
B)−28
C)−2
D)−26
Answer: A — The first differences are −28, −26 and −24, and each is 2 more than the one before.
- B — that is the FIRST difference
- C — the first differences are INCREASING, so the second difference is positive
- D — that is the second of the first differences
1.3
The next TWO terms of 204 ; 176 ; 150 ; 126 ; … are …
(1)
A)104 and 84✓
B)102 and 78
C)104 and 82
D)100 and 74
Answer: A — The first differences continue −22 and then −20.
- B — carried on subtracting 24 each time, ignoring the second difference
- C — used −22 twice instead of −22 then −20
- D — used −26 twice, going backwards through the differences
1.4
For T_n = an² + bn + c, the value of a is …
(1)
A)half the second difference✓
B)the second difference itself
C)the first difference
D)the first term
Answer: A — The second difference always equals 2a.
- B — the second difference is 2a, so a is half of it
- C — the first difference changes from term to term
- D — the first term is a + b + c, not a
1.5
The general term of 204 ; 176 ; 150 ; 126 ; … is …
(1)
A)T_n = n² − 31n + 234✓
B)T_n = 2n² − 31n + 234
C)T_n = n² − 28n + 231
D)T_n = n² + 31n + 234
Answer: A — 2a = 2 gives a = 1; 3a + b = −28 gives b = −31; and a + b + c = 204 gives c = 234.
- B — used a = 2, the second difference itself, instead of half of it
- C — used the first difference −28 as b directly
- D — b must be negative, since the pattern falls at first
1.6
A quadratic pattern with a > 0 has …
(1)
A)a minimum term✓
B)a maximum term
C)no turning term
D)a constant first difference
Answer: A — The terms lie on an upward parabola, so they fall and then rise.
- B — a maximum term needs a < 0
- C — the parabola always turns somewhere
- D — a constant FIRST difference would make the pattern linear
1.7
The general term of 3 ; 8 ; 17 ; 30 ; … is …
(1)
A)T_n = 2n² − n + 2✓
B)T_n = 4n² − n + 2
C)T_n = 2n² + n + 2
D)T_n = 2n² − n
Answer: A — The second difference is 4, so a = 2; then 3a + b = 5 gives b = −1, and c = 2.
- B — used the second difference 4 as a instead of half of it
- C — 3a + b = 5 with a = 2 gives b = −1, not +1
- D — left out c, which would make T₁ equal 1, not 3
1.8
For T_n = n² − 31n + 234, the SMALLEST term occurs at …
(1)
A)n = 15 and n = 16, which are equal✓
B)n = 15,5
C)n = 31
D)n = 1
Answer: A — The vertex is at n = 15,5, and n must be a whole number, so the two neighbouring terms tie at −6.
- B — n counts terms, so it must be a whole number
- C — 31 is the value of −b, before dividing by 2a
- D — the pattern falls at first, so the first term is not the smallest
1.9
For T_n = n² − 31n + 234, the number of NEGATIVE terms is …
(1)
A)4✓
B)5
C)6
D)2
Answer: A — n² − 31n + 234 = (n − 13)(n − 18), which is negative for 13 < n < 18, so n = 14, 15, 16 and 17.
- B — included n = 13 or n = 18, where the term is exactly 0
- C — included both endpoints, but a zero term is not a negative one
- D — counted only the two smallest terms
1.10
The FIRST differences of a quadratic pattern themselves form …
(1)
A)a linear pattern✓
B)a quadratic pattern
C)a geometric pattern
D)a constant pattern
Answer: A — They change by the constant second difference each time, which is what a linear pattern does.
- B — the SECOND differences would then be changing, not constant
- C — a geometric pattern multiplies by a ratio, it does not add
- D — constant first differences would make the ORIGINAL pattern linear

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

[12 MARKS]

A quadratic number pattern T_n = an² + bn + c has these properties:
• T₁ = −2
• the n^th term of the FIRST differences is −6n − 2

2.1
Show that a = −3 and b = 1.
(4)
The first differences of an² + bn + c are a(2n + 1) + b (1)
2an + a + b = −6n − 2 (1)
2a = −6, so a = −3 (1)
a + b = −2 gives b = 1 (1)
2.2
Hence determine the general term T_n.
(3)
T₁ = a + b + c = −2 (1)
−3 + 1 + c = −2, so c = 0 (1)
T_n = −3n² + n (1)
2.3
Determine whether −19 122 is a term of the sequence. Justify your answer with calculations.
(5)
−3n² + n = −19 122 (1)
3n² − n − 19 122 = 0 (1)
Δ = 1 + 4(3)(19 122) = 229 465 (1)
229 465 ≈ 479,03, which is not a whole number (1)
n is not a natural number, so −19 122 is NOT a term (1)

Question 3

[12 MARKS]

Consider the quadratic pattern 3 ; 8 ; 17 ; 30 ; 47 ; …

3.1
Show that the pattern is quadratic and determine its general term.
(5)
First differences 5; 9; 13; 17 (1), second differences 4; 4; 4 — constant (1)
2a = 4 so a = 2 (1); 3a + b = 5 so b = −1 (1); a + b + c = 3 so c = 2
T_n = 2n² − n + 2 (1)
3.2
Determine which term of the pattern is equal to 122.
(4)
2n² − n − 120 = 0 (2)
(n − 8)(2n + 15) = 0 (1)
n = −7,5 is rejected, so it is the 8th term (1)
3.3
Determine the value of the FIRST difference between T₂₀ and T₂₁.
(3)
The first differences have the rule a(2n + 1) + b = 4n + 1 (2)
At n = 20: 81 (1)

TOTAL: 34 marks

This question paper consists of 3 questions.