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Sequences

Arithmetic and geometric sequences and series: general terms, sums, and applied growth problems.

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lessons in this chapter

previous exam questions

4 hr

suggested study time

Hard

chapter difficulty

Last reviewed Sep 2026 · GetCSCA editors

1. Arithmetic Sequences 2. Geometric Sequences

Study order

Work the lessons top to bottom — each ends with practice, a timed set and a 5-stage drill. Then prove it in a full 48Q mock.

Arithmetic Sequences

Lesson 1 · target ~70s per question

Constant-difference sequences: general term, sum of the first n terms, and finding terms from partial sums.

Why CSCA tests this

Arithmetic sequences test whether you can move fluently between a term, the difference, and the paired sum — the template for all CSCA series word problems.

Key points — memorize

Diagnose arithmetic by a constant first difference d = aₙ₊₁ − aₙ; the general term is then linear: aₙ = a₁ + (n − 1)d.
Both sum forms pair first with last: Sₙ = n(a₁ + aₙ)/2 counts n pairs averaging to the midpoint.
Recover hidden terms from partial sums with aₙ = Sₙ − Sₙ₋₁, and a₁ = S₁ directly.
Three-term arithmetic conditions collapse to 2b = a + c, and symmetric triples are best written a − d, a, a + d.
Sₙ as a function of n is quadratic with zero constant term, so a quadratic-with-constant partial sum signals non-arithmetic from n = 1.

Formula sheet

aₙ = a₁ + (n − 1)d
Sₙ = n(a₁ + aₙ)/2
Sₙ = n(2a₁ + (n − 1)d)/2
aₙ = Sₙ − Sₙ₋₁ (n ≥ 2)
a, b, c in A.P. ⇔ 2b = a + c
d = aₙ − aₙ₋₁; aₖ = aₘ + (k − m)d
1 + 2 + … + n = n(n + 1)/2
Sₙ is quadratic in n (no constant term); aₙ is linear in n

Classic traps

Off-by-one indexing: using n instead of n − 1 in a₁ + (n−1)d, especially for 'the 10th term'.
Applying aₙ = Sₙ − Sₙ₋₁ at n = 1, where S₀ does not exist (use a₁ = S₁).
Assuming any quadratic Sₙ means arithmetic — the constant term must be zero.
Mixing up aₙ (one term) with Sₙ (sum of n terms) in word problems.

Exam tactic

Write down a₁, d, n explicitly before touching a formula — half of arithmetic errors are plugging the wrong n.

Worked examples

Example 1: The 1st term is 3 and the common difference is 4. Find the 10th term.
1. a₁₀ = 3 + (10 − 1) · 4 = 3 + 36.
Answer: 39
Example 2: Find the sum 1 + 2 + … + 100.
1. Arithmetic with a₁ = 1, a₁₀₀ = 100, n = 100.
2. S = 100 · (1 + 100)/2 = 50 · 101.
Answer: 5050

FoundationPractice →BasicPractice →CSCA StandardPractice →Timed set · ~70s/Q →

Geometric Sequences

Lesson 2 · target ~70s per question

Constant-ratio sequences: general term, finite and infinite sums, and compound-growth applications.

Why CSCA tests this

Geometric sequences model every CSCA growth story — compound interest, decay, half-life — and the infinite-sum convergence test is a classic trap item.

Key points — memorize

Diagnose geometric by a constant ratio q = aₙ₊₁/aₙ with nonzero terms; the general term is exponential in n: aₙ = a₁qⁿ⁻¹.
Finite sums multiply out the (1 − q) factor: Sₙ = a₁(1 − qⁿ)/(1 − q) for q ≠ 1, and Sₙ = na₁ when q = 1.
The infinite sum S = a₁/(1 − q) exists if and only if |q| < 1; verify the ratio bound before using it.
Three-term geometric conditions collapse to b² = ac (same sign required for real progressions).
Translate growth/decay language directly: 'r% per period' means q = 1 ± r, and repeated multiplication is a geometric sequence.

Formula sheet

aₙ = a₁ · qⁿ⁻¹
Sₙ = a₁(1 − qⁿ)/(1 − q), q ≠ 1
S = a₁/(1 − q), |q| < 1
a, b, c in G.P. ⇔ b² = ac
Compound growth: A = P(1 + r)ⁿ
q = aₙ/aₙ₋₁; aₖ = aₘ · q^(k−m)
Sₙ = a₁(qⁿ − 1)/(q − 1), q ≠ 1 (equivalent form)
|q| ≥ 1 ⇒ infinite series diverges (no finite sum)

Classic traps

Using the infinite-sum formula when |q| ≥ 1, where the series diverges.
Off-by-one exponents: writing aₙ = a₁qⁿ instead of a₁qⁿ⁻¹.
Dividing by (1 − q) without handling q = 1 as the separate Sₙ = na₁ case.
Confusing arithmetic difference with geometric ratio in mixed word problems.

Exam tactic

Check |q| < 1 first on any 'infinite sum' item — if it fails, the answer is 'diverges', no computation needed.

Worked examples

Example 1: Find the 6th term of 2, 6, 18, ….
1. Ratio q = 6/2 = 3, a₁ = 2.
2. a₆ = 2 · 3⁵ = 2 · 243.
Answer: 486
Example 2: Find 1 + 1/2 + 1/4 + … (infinite sum).
1. Geometric with a₁ = 1, q = 1/2, |q| < 1.
2. S = 1/(1 − 1/2) = 2.
Answer: 2

FoundationPractice →BasicPractice →CSCA StandardPractice →Timed set · ~70s/Q →

Previous exam questions

Real CSCA-style questions tagged to Sequences — answer right here.

20 previous exam questions · 0/0 correct

Dec 2025 · Q5arithmetic-sequences
Arithmetic $a_1 = 2$ , $d = 3$ : then $a_{100} =$
Dec 2025 · Q13arithmetic-sequences
$a, b, c$ arithmetic with $a + c = 20$ : then $b =$
Dec 2025 · Q17arithmetic-sequences
Arithmetic $a_1 = 1, a_2 = 2$ : then $a_{2025} =$
Dec 2025 · Q23geometric-sequences
Geometric 1, 2, 4: general term is
Dec 2025 · Q41recursive-sequences
$a_1 = 1$ , $a_n = \frac{1}{1+\frac{1}{a_{n-1}}}$ : $a_{100} =$
Dec 2025 · Q47arithmetic-sequences
Arithmetic: $3(a_2+a_6)+2(a_6+a_{10}+a_{14}) = 24$ : $S_{13} =$
Jan 2026 · Q4arithmetic-sequences
Arithmetic $a_1 = 2$ , $d = 3$ : $a_{100} =$
Jan 2026 · Q12arithmetic-sequences
Arithmetic mean of $2-\sqrt{3}$ and $2+\sqrt{3}$ is

Related guides

30-day study plan → fit this chapter into a week-by-week system
75-second pacing rule → hold 75s/question once the content clicks
Top 10 mistakes → the error patterns that cost the most points

2 lessons · one chapter

Finished Sequences? Prove it.

Run a timed set, clear every mistake, then take a full 48Q mock.

Practice questions →Full mock →