Grade 11 Maths · Number Patterns, Sequences and Series

Find a quadratic general term

Write the formula Tn = an² + bn + c for a pattern whose second differences are equal, for example 2, 5, 10, 17 gives Tn = n² + 1.

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Quadratic number patterns grade 11: introduction and examplesKevinmathscience

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Question 3.1 · May/June 2025 · Paper 1 · National (DBE) · 3 marks

Given the quadratic sequence:

14; 9; 6; 5; ...

3.1 Show that the general term of this sequence is

T_{n} = n^{2} - 8 n + 21

.(3)

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More past paper questions: find a quadratic general term

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3. Question 3.1.1 · May/June 2024 · Paper 1 · National (DBE) · 3 marks
3.1 Consider the quadratic number pattern: $3; 7; 12; \dots$
3.1.1 Show that the general term of this number pattern is given by $T_{n} = \frac{1}{2} n^{2} + \frac{5}{2} n$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Determine second difference $2 a = 1 ⟹ a = 0, 5$
  1 mark · Find a quadratic general term
2. ✓
  Solve for b using first difference $3 a + b = 4 ⟹ b = 2, 5$
  1 mark · Find a quadratic general term
3. ✓
  Solve for c using first term $a + b + c = 3 ⟹ c = 0$
  1 mark · Find a quadratic general term
4. Question 2.2.2 · November 2024 · Paper 1 · National (DBE) · 5 marks
2.2 The sequence of the first differences of a quadratic pattern is: $1; 3; 5; ...$
2.2.2 If it is further given that the third term of the quadratic pattern is $32$ , determine the general term, $T_{n}$ , of the quadratic pattern.(5)
Show the memo (4 ticks)Memo
1. ✓
  Find second difference (2a)Second difference is constant: $3 - 1 = 2$
  $2 a = 2 ⟹ a = 1$
  1 mark · Find a quadratic general term
2. ✓
  Calculate b $3 a + b = 1$
  $3 (1) + b = 1$
  $b = - 2$
  1 mark · Find a quadratic general term
3. ✓
  Calculate c using T3 $T_{n} = n^{2} - 2 n + c$
  $T_{3} = (3)^{2} - 2 (3) + c = 32$
  $9 - 6 + c = 32$
  $3 + c = 32$
  $c = 29$
  2 marks · Find a quadratic general term
4. ✓
  Final general term $T_{n} = n^{2} - 2 n + 29$
  1 mark · Find a quadratic general term
5. Question 2.1.1 · March 2025 · Provincial · 4 marks
The given number pattern is a combination of a quadratic sequence and an arithmetic sequence: $16; 32; 0; 28; - 12; 24; - 20; 20; ...$
2.1 Pattern analysis questions.
2.1.1 Determine the general term of the quadratic sequence.(4)
Show the memo (5 ticks)Memo
1. ✓
  Extract quadratic sequence and find differencesSequence: $16; 0; - 12; - 20$
  First differences: $- 16; - 12; - 8$
  Second difference: $4$
  0 marks · Find a quadratic general term
2. ✓
  Calculate coefficient a $2 a = 4$
  $a = 2$
  1 mark · Find a quadratic general term
3. ✓
  Calculate coefficient b $3 a + b = - 16$
  $3 (2) + b = - 16$
  $b = - 22$
  1 mark · Find a quadratic general term
4. ✓
  Calculate coefficient c $a + b + c = 16$
  $2 - 22 + c = 16$
  $c = 36$
  1 mark · Find a quadratic general term
5. ✓
  Write final general term $T_{n} = 2 n^{2} - 22 n + 36$
  1 mark · Find a quadratic general term
6. Question 2.1.1 · March 2024 · Paper 1 · Provincial · 4 marks
2.1 The given number pattern is a combination of a quadratic sequence and an arithmetic sequence: $16; 32; 0; 28; - 12; 24; - 20; 20; \dots$
2.1.1 Determine the general term of the quadratic sequence.(4)
Show the memo (4 ticks)Memo
1. ✓
  Extract quadratic sequence and find differencesSequence: $16; 0; - 12; - 20; \dots$
  First differences: $- 16; - 12; - 8$
  Second difference: $4$
  1 mark · Find a quadratic general term
2. ✓
  Solve for b $3 a + b = - 16$
  $3 (2) + b = - 16$
  $b = - 22$
  1 mark · Find a quadratic general term
3. ✓
  Solve for c $a + b + c = 16$
  $2 - 22 + c = 16$
  $c = 36$
  1 mark · Find a quadratic general term
4. ✓
  Write final general term $T_{n} = 2 n^{2} - 22 n + 36$
  1 mark · Find a quadratic general term

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