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Patterns

Quadratic number patterns

First and second differences. Tₙ = an² + bn + c with 2a = second difference.

CAPS

Grade 11 · Mathematics · Patterns

This sits on Mathematics Paper 1.

  • The paper

    Paper 1 is algebra, exponents, equations, number patterns, functions, finance and probability. Copy the formula, then substitute.

  • Cognitive level

    CAPS spreads the marks across knowledge, routine procedures, complex procedures and problem solving. A routine question still wants the line of working. A problem-solving question wants you to choose the method, not only to calculate.

  • The relation

    2a = second difference. Leading coefficient

  • How a marker reads it

    Always show the difference table. Method marks sit in the second-difference row.

Each Mathematics paper is 150 marks. A correct substitution can earn a method mark even when the last arithmetic slips, if the formula was the right one.

-2-1012

Diagram

Worked example

Sequence 3; 8; 15; 24; … Find Tₙ and T₁₀.

Before any working — hunt first

Given

Numbers and units already in the sentence: 3 · 8 · 15 · 24. Copy each one into a column. A name without a number is not given.

Missing

After you copy the given, the letter that still has no number is missing. The paper is pointing at: Tₙ and T₁₀. Circle that word — that is the missing variable.

Find

We are required to find: Tₙ and T₁₀. Write that as a symbol on the left, with a question mark, before you touch a formula.

Calculate

Write 2a = second difference with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Tₙ and T₁₀.

Simplified approach

First differences 5, 7, 9. Second differences constantly 2, so a = 1. Fit b and c with n = 1 and n = 2.

Culture note

A Limpopo mealie-crib that adds one extra row of cobs each season grows as a quadratic — the extra-extra is the second difference.

Formulas

2a=second difference

Leading coefficient

Tₙ=an²+bn+c

General term

How to hunt the missing letter

Do not start by naming the letters. First hunt the missing one: copy every number you can see (Given). The letter with no number is Missing. The asking word is Find. The formula that still contains that letter is what you Calculate.

How to hunt: (1) Underline every number and its unit — those are Given. (2) Circle the asking word — find, calculate, how far, what is. That word names Find. (3) The letter the asking word names, which has no number yet, is Missing. (4) Write the formula with empty letters. Cross out every given letter. The letter still standing is what you Calculate.

  • a

    second-difference / 2a

    How you spot this in the sentence

    When the sentence names the second-difference / 2a, that letter is a. Copy the number next to it into Given. If the sentence never names it, a is Missing. In Tₙ = an² + bn + c, the constant second difference equals 2a.

    Example:If second differences are 4, then 2a = 4 so a = 2.

  • b

    coefficient of x

    How you spot this in the sentence

    When the sentence names the coefficient of x, that letter is b. Copy the number next to it into Given. If the sentence never names it, b is Missing. The linear piece of Tₙ = an² + bn + c. Find it from first terms after you know a.

    Example:Once a is known, use T₁ and T₂ to solve for b and c.

  • c

    constant term

    How you spot this in the sentence

    When the sentence names the constant term, that letter is c. Copy the number next to it into Given. If the sentence never names it, c is Missing. Tₙ when n = 0 in the formula, or T₁ = a + b + c.

    Example:If T₁ = 3 and a = 2, b = −1, then c = 3 − 2 − (−1) = 2.

  • Tₙ

    nth term

    How you spot this in the sentence

    When the sentence names the nth term, that letter is Tₙ. Copy the number next to it into Given. If the sentence never names it, Tₙ is Missing. The general term of the quadratic pattern.

    Example:Tₙ=2n²−n+2, so T₅=50−5+2=47.

Exam tricks you must know

These are the moves markers look for — not shortcuts that skip the method, but the habits that earn the method mark.

  • Always CAST — the second quadrant is a mark on its own

    How:Sketch CAST. Read the sign of sin, cos or tan from the quadrant, not from memory of a special angle. For a general solution write the acute angle first, then 180° − θ or 360° − θ. Leaving the second angle is the KZN and Gauteng dock.

    When to use it:Any sin / cos / tan equation that asks for θ in an interval.

  • Decide first: is money going in (F) or a loan being paid off (P)?

    How:Underline the asking word. “Future value”, “how much in the account”, “accumulate” → F. “Present value”, “loan”, “outstanding balance”, “how much can she borrow” → P. Write F or P in the Find cell before you open the formula sheet. The wrong letter loses the whole question.

    When to use it:Every compound-interest and annuity question on Paper 1.

  • Ask: does ABC differ from ACB? Yes → permutation. No → combination

    How:Read the noun. Arrangements, passwords, in a line, first/second/third → ⁿPᵣ. Committees, teams, groups, chosen → ⁿCᵣ. Write that sentence as the method line. Then substitute n and r.

    When to use it:Counting, arrangements, and probability “how many ways” questions.

Step 1 / 7

Differences

Δ1: 5, 7, 9. Δ2: 2, 2. So 2a=2→a=1.

a=1

Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.

4. Step-by-step working

  1. 01

    Differences

    Δ1: 5, 7, 9. Δ2: 2, 2. So 2a=2→a=1.

    a=1

  2. 02

    Fit n = 1

    T₁=a+b+c=3→1+b+c=3→b+c=2.

  3. 03

    Fit n = 2

    T₂=4+2b+c=8→2b+c=4. Subtract: b=2, c=0.

    Tₙ=n²+2n

  4. 04

    T₁₀

    T₁₀=100+20=120.

    120

  5. 05

    Step F — Box the answer with its unit

    Tₙ=n²+2n; T₁₀=120

More worked examples

Board walkthroughs on this topic. Open one, copy the working, then try the next.

Example 1 / 20

Tₙ=n²+n. Find T₅.

4. Step-by-step working

  1. 01

    nth term

    Substitute n=5.

  2. 02

    Calculate

    25+5=30

    30

How we got there

Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.

Explain in another language · 11

1. What is the question asking?

Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Tₙ = n² + n. Find T₅.

Before any working — hunt first

Given

Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.

Missing

After you copy the given, the letter that still has no number is missing. The paper is pointing at: T₅. Circle that word — that is the missing variable.

Find

We are required to find: T₅. Write that as a symbol on the left, with a question mark, before you touch a formula.

Calculate

Write 30 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives T₅.

In other words — as if you have never seen this before

If you have never seen this before: Substitute n into the general term, or use the second difference (2a). We will name a method in words, write the formula with no numbers, then substitute one line at a time.

2. What do we already know?

Tₙ=n²+n, n=5.

3. Which method — and why this one?

Substitute n into the general term, or use the second difference (2a).

4. Step-by-step working

  1. 01 / 05

    nth term

    Substitute n=5.

    Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.

  2. 02 / 05

    Calculate

    25+5=30

    30

    Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.

  3. 03 / 05

    Step G — Sense-check the size

    Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.

    Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.

  4. 04 / 05

    Step F — Box the answer with its unit

    30

    Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.

  5. 05 / 05

    Step H — What the marker is looking for

    On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.

    Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.

5. The answer

30

Copy onto your answer sheet — in this order

Formula: 30

Answer: 30

6. Check it like a marker

Tₙ is the term in position n, not n itself. For a quadratic pattern the second difference equals 2a, not a.

7. A trap first-timers fall into

First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”

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