Both sequence types and the Grade 12 exam questions on them — finding a term, finding which term equals a value, the sum of a series, and the sum to infinity — each worked in full, with a worksheet.
A sequence is an ordered list of numbers. In this lesson you work with two of the Grade 12 sequence types — arithmetic and geometric — and the skill is recognising which one you have, then applying the right formula. It covers both, plus the series (sum) formulas exams love. Grade 12 also opens by revising Grade 11's quadratic number patterns (a constant second difference); those have their own lesson, Quadratic Number Patterns (Grade 11 & 12).
1The formulas (all on your data sheet)
a = first term, d = common difference (you add), r = common ratio (you multiply).
2Type 1: Identify the type and find a term
- 1Test: 8 − 5 = 3, 11 − 8 = 3 → arithmetic, d = 3, a = 5.Constant difference means arithmetic.
- 2Tₙ = a + (n − 1)d → T₁₀ = 5 + 9(3).Substitute n = 10.
- 3= 5 + 27 = 32.
- 1Test: 6 ÷ 2 = 3, 18 ÷ 6 = 3 → geometric, r = 3, a = 2.Constant ratio means geometric.
- 2Tₙ = a·rn − 1 → T₆ = 2·35.Substitute n = 6.
- 3= 2 × 243 = 486.
3Type 2: Which term equals a given value?
- 1Set Tₙ = 92: 5 + (n − 1)3 = 92.You know the value, want the position n.
- 2(n − 1)3 = 87 → n − 1 = 29 → n = 30.Solve the equation.
- 3So 92 is the 30th term.n came out a whole number, so 92 is in the sequence.
4Type 3: Sum of a series (Sₙ)
- 1Arithmetic, a = 5, d = 3, n = 20.Use the arithmetic sum formula.
- 2S₂₀ = 20/2 [2(5) + 19(3)] = 10[10 + 57].Substitute into Sₙ = n/2[2a + (n − 1)d].
- 3= 10 × 67 = 670.
5Type 4: Sum to infinity (geometric, |r| < 1)
When a geometric series has −1 < r < 1, the terms shrink and the total settles on a finite value:
- 1Geometric: r = 4/8 = 1/2 (|r| < 1, so it converges).Sum to infinity only exists when |r| < 1.
- 2S∞ = a/(1 − r) = 8/(1 − 1/2) = 8/(1/2).a = 8.
- 3= 16.The running total gets ever closer to 16.
First test which type you have: subtract consecutive terms (constant difference → arithmetic) or divide them (constant ratio → geometric). Using the wrong formula is the main way marks are lost.
It's n − 1, not n. The term formulas use the power/multiplier n − 1. Forgetting the '− 1' shifts every term by one position.
Sequences and series are among the most predictable marks in the NSC Paper 1 — the formulas are given, so the skill is choosing the right one and substituting carefully. Practise every type below.
Practice exercises
Work each one out, then click to reveal the answer.
- 110th term of 3 ; 7 ; 11 ; … (arithmetic term)
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a = 3, d = 4; T₁₀ = 3 + 9(4) = 39 - 28th term of 2 ; 5 ; 8 ; … (arithmetic term)
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a = 2, d = 3; T₈ = 2 + 7(3) = 23 - 35th term of 3 ; 6 ; 12 ; … (geometric term)
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a = 3, r = 2; T₅ = 3·24 = 48 - 46th term of 1 ; 3 ; 9 ; … (geometric term)
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a = 1, r = 3; T₆ = 35 = 243 - 5Which term of 4 ; 7 ; 10 ; … is 61? (which term)
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4 + (n − 1)3 = 61 → n = 20 - 6Which term of 2 ; 4 ; 8 ; … is 256? (which term)
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2·2n − 1 = 256 → 2n = 256 → n = 8 - 7Sum of the first 10 terms of 2 ; 4 ; 6 ; … (arithmetic sum)
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S₁₀ = 10/2[2(2) + 9(2)] = 5(22) = 110 - 8Sum of the first 15 terms of 3 ; 5 ; 7 ; … (arithmetic sum)
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S₁₅ = 15/2[6 + 14(2)] = 15/2(34) = 255 - 9Sum to infinity of 6 + 3 + 1,5 + … (S∞)
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r = 1/2; 6/(1 − 1/2) = 12 - 10Sum to infinity of 10 + 2 + 0,4 + … (S∞)
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r = 0,2; 10/(1 − 0,2) = 10/0,8 = 12,5
Quick Quiz
5 quick questions on what you just read. Take it when you feel ready.
Now practise it
Download Grade 12 past papers and worksheets on this topic.