Sequences & series

Arithmetic & geometric series

Sₙ for arithmetic and geometric, infinite geometric |r| < 1, and sigma notation.

CAPS

Grade 12 · Mathematics · Sequences & series

This sits on Mathematics Paper 1.

The paper
Paper 1 is algebra, sequences and series, finance, functions, differential calculus and probability. If a formula is on the formula sheet, copy it before you 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
Tₙ = a rⁿ⁻¹. Geometric nth term — exponent is n minus 1
How a marker reads it
If |r| ≥ 1, write “the series diverges” — do not force the formula.

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.

Diagram

Worked example

Find the sum to infinity of 8 + 4 + 2 + 1 + …

Before any working — hunt first

Given

Numbers and units already in the sentence: 8 · 4 · 2 · 1. 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: the sum to infinity of 8 + 4 + 2 + 1 + …. Circle that word — that is the missing variable.

Find

We are required to find: the sum to infinity of 8 + 4 + 2 + 1 + …. Write that as a symbol on the left, with a question mark, before you touch a formula.

Calculate

Write Tₙ = a rⁿ⁻¹ with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives the sum to infinity of 8 + 4 + 2 + 1 + ….

Simplified approach

First term a = 8, r = 12. |r| < 1 so the infinite sum exists: a(1−r).

Culture note

Halving a loaf at a family table in Soweto, again and again, still adds up to one whole loaf — infinite geometric.

Formulas

Tₙ=a rⁿ⁻¹

Geometric nth term — exponent is n minus 1

Sₙ=n2 [2a + (n−1)d]

Arithmetic sum

Sₙ=a(rⁿ − 1)(r − 1), r ≠ 1

Geometric sum

S_∞=a(1 − r), |r| < 1

Infinite geometric

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
first term
How you spot this in the sentence
When the sentence names the first term, that letter is a. Copy the number next to it into Given. If the sentence never names it, a is Missing. The first number of the sequence — T₁. Same letter for arithmetic and geometric.
Example:In 3; 6; 12; 24; …, a is 3. In 5; 8; 11; …, a is 5.
r
common ratio
How you spot this in the sentence
When the sentence names the common ratio, that letter is r. Copy the number next to it into Given. If the sentence never names it, r is Missing. Geometric only: the number you multiply by each time. r = T₂ / T₁ = T₃ / T₂. Never subtract terms to find r.
Example:In 3; 6; 12; 24; …, r = 63=2. In 8; 4; 2; 1; …, r = 48 = ½.
d
common difference
How you spot this in the sentence
When the sentence names the common difference, that letter is d. Copy the number next to it into Given. If the sentence never names it, d is Missing. Arithmetic only: what you add each time. d = T₂ − T₁. Do not use d on a geometric sequence — use r.
Example:In 5; 8; 11; …, d = 3. 3; 6; 12 multiplies, so d does not apply.
n
term number
How you spot this in the sentence
When the sentence names the term number, that letter is n. Copy the number next to it into Given. If the sentence never names it, n is Missing. The position of the term you want, or how many terms you add in Sₙ. Counting starts at n = 1.
Example:T₅ is the 5th term (n = 5). S₁₀ adds the first 10 terms (n = 10).
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. Geometric: Tₙ = a rⁿ⁻¹. The exponent is n minus 1, not n. Arithmetic uses Tₙ = a + (n − 1)d.
Example:With a = 3, r = 2, n = 5: T₅ = 3 × 2⁴ = 3 × 16 = 48.
Sₙ
sum of n terms
How you spot this in the sentence
When the sentence names the sum of n terms, that letter is Sₙ. Copy the number next to it into Given. If the sentence never names it, Sₙ is Missing. Geometric: Sₙ = a(rⁿ − 1)/(r − 1) when r ≠ 1. Arithmetic: Sₙ = n/2 [2a + (n − 1)d].
Example:a=3, r=2, n=4: S₄=3(16 − 1)(2 − 1)=45. Check: 3+6+12+24=45.
S_∞
sum to infinity
How you spot this in the sentence
When the sentence names the sum to infinity, that letter is S_∞. Copy the number next to it into Given. If the sentence never names it, S_∞ is Missing. Only a geometric series with |r| < 1. S_∞ = a / (1 − r). If |r| ≥ 1 the series diverges — do not force the formula.
Example:8+4+2+1+… has a=8, r=½. S_∞=8(1 − ½)=16.
|r|
absolute value of r
How you spot this in the sentence
When the sentence names the absolute value of r, that letter is |r|. Copy the number next to it into Given. If the sentence never names it, |r| is Missing. How far r sits from 0. S_∞ exists only when |r| < 1, i.e. −1 < r < 1. r = ±1 is not allowed.
Example:r = ½ → |r| = ½ < 1, converges. r = −⅓ → |r| = ⅓ < 1, converges. r = 2 → |r| = 2 ≥ 1, diverges.

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.

If |r| ≥ 1, write “the series diverges” — do not force S_∞
How:Hunt r first: r = T₂ / T₁. Then test |r| < 1 before you touch S_∞ = a(1 − r). If the test fails, the Find cell is not a number — it is the sentence “diverges”. Forcing the formula is a method-mark zero.
When to use it:Infinite geometric series and “sum to infinity” wording.
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.

Step 1 / 9

Sequence, then series

8; 4; 2; 1; … is the geometric sequence (the list). Adding them — 8 + 4 + 2 + 1 + … — is the series. The dots mean it never stops, so we want S_∞, the number the totals climb towards.

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

01
Name a and r
a is the first term: a = 8. r is what you multiply by: r = 48=12. Check the next pair: 24=12 — same r, so it is geometric.
a=8,r=12
02
The gate: |r| < 1
|r| is how far r sits from 0. |12| = 12, which is less than 1, so the terms shrink and S_∞ exists. If |r| ≥ 1 you stop and write “the series diverges” — do not force the formula.
|r|=12 < 1
03
Why the leftover vanishes
The finite sum is Sₙ = a(1 − rⁿ)/(1 − r). As n gets huge, (1/2)ⁿ → 0, so the rⁿ piece dies and you are left with S_∞ = a(1 − r).
S_∞=a(1 − r)
04
Substitute
S_∞=8(1 − 1/2)=8(1/2)=8×2=16.
S_∞=16
05
The trap: |r| ≥ 1
2 + 4 + 8 + 16 + … has r = 2. |2| ≥ 1, so the series diverges. Writing a(1−r)=2(1−2)=−2 is nonsense — the totals 2, 6, 14, 30, … explode, they do not go to −2.
diverges

More worked examples

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

Example 1 / 20

AP: a=1, d=1, n=10. Find Sₙ.

4. Step-by-step working

01
Sum formula
Sₙ=n2 [2a + (n − 1)d].
Sₙ=n2 [2a+(n−1)d]
02
Substitute
S¹⁰=102 [2(1) + (10−1)(1)]=55.
S¹⁰=55

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: AP: a = 1, d = 1, n = 10. Find Sₙ.

Before any working — hunt first

Given

Numbers and units already in the sentence: 1 · 1 · 10. 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: Sₙ. Circle that word — that is the missing variable.

Find

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

Calculate

Write Sₙ = n2 [2a+(n−1)d] with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Sₙ.

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

If you have never seen this before: Sₙ = n2 [2a + (n−1)d]. 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?

a=1, d=1, n=10.

3. Which method — and why this one?

Sₙ=n2 [2a + (n−1)d].

4. Step-by-step working

01 / 05
Sum formula
Sₙ=n2 [2a + (n − 1)d].
Sₙ=n2 [2a+(n−1)d]
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.
02 / 05
Substitute
S¹⁰=102 [2(1) + (10−1)(1)]=55.
S¹⁰=55
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.
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.
04 / 05
Step F — Box the answer with its unit
55
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.
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

Copy onto your answer sheet — in this order

Formula: Sₙ=n2 [2a+(n−1)d]

Answer: 55

6. Check it like a marker

Last term Tₙ=1+(10-1)1=10. Sₙ=n2 (a + last).

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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