What you'll learn
This revision guide covers all testable content on sequences and series for CXC CSEC Additional Mathematics. You will master arithmetic and geometric progressions, including finding general terms, calculating sums to n terms, working with sigma notation, and solving problems involving infinite geometric series. The content aligns precisely with the CXC syllabus requirements.
Key terms and definitions
Sequence — an ordered list of numbers following a specific pattern or rule, where each number is called a term.
Series — the sum of the terms of a sequence, expressed as the addition of consecutive terms.
Arithmetic progression (AP) — a sequence where each term differs from the previous term by a constant value called the common difference (d).
Geometric progression (GP) — a sequence where each term is obtained by multiplying the previous term by a constant value called the common ratio (r).
Common difference (d) — the constant value added to each term in an arithmetic progression to obtain the next term, calculated as d = u₂ - u₁.
Common ratio (r) — the constant value by which each term in a geometric progression is multiplied to obtain the next term, calculated as r = u₂/u₁.
Convergent series — an infinite geometric series where |r| < 1, allowing the sum to approach a finite value.
Sigma notation (Σ) — a mathematical shorthand for expressing the sum of a series using the Greek capital letter sigma.
Core concepts
Arithmetic progressions
An arithmetic progression has the general form: a, a + d, a + 2d, a + 3d, ...
where a is the first term and d is the common difference.
The nth term formula: uₙ = a + (n - 1)d
Sum of the first n terms: Sₙ = n/2[2a + (n - 1)d]
Alternative form: Sₙ = n/2(a + l)
where l is the last term.
Key applications:
- Finding any term when given the first term and common difference
- Determining how many terms sum to a particular value
- Solving problems involving the position of specific terms
Example context: A banana plantation in Jamaica increases production by 250 boxes each month. If production starts at 1200 boxes in January, this forms an AP with a = 1200 and d = 250.
Geometric progressions
A geometric progression has the general form: a, ar, ar², ar³, ...
where a is the first term and r is the common ratio.
The nth term formula: uₙ = arⁿ⁻¹
Sum of the first n terms: Sₙ = a(1 - rⁿ)/(1 - r) when r ≠ 1
Alternative form: Sₙ = a(rⁿ - 1)/(r - 1) when r ≠ 1
Sum to infinity: When |r| < 1, the infinite series converges: S∞ = a/(1 - r)
Key applications:
- Compound interest problems in Caribbean financial contexts
- Population growth models
- Depreciation calculations for equipment or vehicles
- Problems involving repeated percentage changes
Example context: A fishing boat in Barbados depreciates by 15% annually. Starting at $80,000, the values form a GP with a = 80,000 and r = 0.85.
Identifying sequence types
For arithmetic progressions:
- Check if consecutive differences are constant
- Calculate: u₂ - u₁, u₃ - u₂, u₄ - u₃
- If equal, the sequence is arithmetic
For geometric progressions:
- Check if consecutive ratios are constant
- Calculate: u₂/u₁, u₃/u₂, u₄/u₃
- If equal, the sequence is geometric
Neither AP nor GP: Some sequences follow other patterns (quadratic sequences, Fibonacci-type sequences). These may require:
- Finding differences of differences
- Pattern recognition
- Using general term formulas provided in questions
Sigma notation
Sigma notation expresses series compactly:
Σᵢ₌₁ⁿ uᵢ means u₁ + u₂ + u₃ + ... + uₙ
Standard results:
- Σᵢ₌₁ⁿ c = nc (where c is constant)
- Σᵢ₌₁ⁿ i = n(n + 1)/2
- Σᵢ₌₁ⁿ i² = n(n + 1)(2n + 1)/6
- Σᵢ₌₁ⁿ i³ = [n(n + 1)/2]²
Properties of sigma notation:
- Σ(auᵢ + bvᵢ) = aΣuᵢ + bΣvᵢ
- Constants can be factored out: Σcuᵢ = cΣuᵢ
Working with sigma notation:
- Identify the general term
- Substitute the lower and upper limits
- Apply standard formulas where applicable
- Simplify algebraically
Applications and problem-solving
Financial contexts:
- Savings plans with regular deposits (AP)
- Compound interest investments (GP)
- Loan repayments with reducing balances
- Depreciation schedules
Practical Caribbean scenarios:
- Agricultural production increases
- Tourism visitor growth rates
- Hurricane preparedness funding allocations
- Fisheries quota distributions
Problem-solving strategy:
- Identify whether the situation involves AP or GP
- Extract values for a, d (or r), and n
- Determine what is being asked (term value, sum, number of terms)
- Select the appropriate formula
- Substitute values and solve
- Check reasonableness of answer in context
Mixed problems and advanced applications
Finding unknown terms: When given non-consecutive terms, form simultaneous equations using the general term formulas.
Sum of specific ranges: To find the sum from term p to term q: Sum = Sᵧ - Sₚ₋₁
Geometric means: The geometric mean between two numbers a and b is √(ab). This extends to multiple geometric means inserted between two terms.
Arithmetic means: The arithmetic mean between two numbers a and b is (a + b)/2. Multiple arithmetic means can be inserted by dividing the total difference equally.
Conditions for convergence: An infinite GP converges only when -1 < r < 1. If |r| ≥ 1, the series diverges and has no finite sum.
Worked examples
Example 1: Arithmetic progression application
The Trinidad and Tobago Carnival committee plans seating in rows. The first row has 24 seats, and each subsequent row has 3 more seats than the previous row.
(a) Find the number of seats in the 18th row. (2 marks)
(b) Calculate the total number of seats in the first 18 rows. (2 marks)
(c) Determine which row first contains more than 80 seats. (3 marks)
Solution:
(a) This is an AP with a = 24, d = 3, n = 18
Using uₙ = a + (n - 1)d u₁₈ = 24 + (18 - 1)(3) u₁₈ = 24 + 51 u₁₈ = 75 seats
(b) Using Sₙ = n/2[2a + (n - 1)d] S₁₈ = 18/2[2(24) + (18 - 1)(3)] S₁₈ = 9[48 + 51] S₁₈ = 9(99) S₁₈ = 891 seats
(c) Need uₙ > 80 24 + (n - 1)(3) > 80 (n - 1)(3) > 56 n - 1 > 18.67 n > 19.67
Therefore, the 20th row is the first to contain more than 80 seats.
Example 2: Geometric progression with sum to infinity
A ball is dropped from a height of 8 m. After each bounce, it rises to 3/4 of its previous height.
(a) Write down the heights reached after the 1st, 2nd, and 3rd bounces. (2 marks)
(b) Find the height reached after the 8th bounce. (2 marks)
(c) Calculate the total vertical distance travelled by the ball when it comes to rest. (4 marks)
Solution:
(a) After 1st bounce: 8 × 3/4 = 6 m After 2nd bounce: 6 × 3/4 = 4.5 m After 3rd bounce: 4.5 × 3/4 = 3.375 m
(b) This is a GP with a = 6, r = 3/4 (starting after first bounce) u₈ = 6(3/4)⁷ u₈ = 6(0.1335...) u₈ = 0.80 m (2 d.p.)
(c) Total distance = initial drop + 2(sum of bounce heights) The ball goes up and down each bounce height except the initial drop.
Sum of all bounce heights = a/(1 - r) = 6/(1 - 3/4) = 6/(1/4) = 24 m
Total distance = 8 + 2(24) = 8 + 48 = 56 m
Example 3: Sigma notation
(a) Evaluate Σᵢ₌₁²⁰ (3i - 5). (3 marks)
(b) Find the value of n for which Σᵢ₌₁ⁿ (4i + 1) = 630. (4 marks)
Solution:
(a) Σᵢ₌₁²⁰ (3i - 5) = Σᵢ₌₁²⁰ 3i - Σᵢ₌₁²⁰ 5
= 3Σᵢ₌₁²⁰ i - 20(5)
= 3[20(21)/2] - 100
= 3(210) - 100
= 630 - 100 = 530
(b) Σᵢ₌₁ⁿ (4i + 1) = 4Σᵢ₌₁ⁿ i + Σᵢ₌₁ⁿ 1
= 4[n(n + 1)/2] + n
= 2n(n + 1) + n
= 2n² + 2n + n
= 2n² + 3n
Given: 2n² + 3n = 630
2n² + 3n - 630 = 0
Using the quadratic formula: n = [-3 ± √(9 + 5040)]/4
n = [-3 ± √5049]/4 = [-3 ± 71.06]/4
n = 17 (taking positive value only)
Common mistakes and how to avoid them
Confusing AP and GP formulas: Always identify the sequence type first by checking differences (AP) or ratios (GP) before selecting formulas. Write down which type you're dealing with.
Incorrect sigma notation expansion: Remember that Σᵢ₌₃¹⁰ uᵢ starts at i = 3, not i = 1. Calculate S₁₀ - S₂ when finding partial sums, not S₁₀ alone.
Sign errors in GP sum formula: When r < 1, use Sₙ = a(1 - rⁿ)/(1 - r). When r > 1, use Sₙ = a(rⁿ - 1)/(r - 1) to avoid negative denominators. Both are correct but choosing wisely reduces errors.
Forgetting the n - 1 in geometric terms: The nth term is arⁿ⁻¹, not arⁿ. The first term (n = 1) should give a(r⁰) = a, not ar.
Misapplying sum to infinity: Only use S∞ = a/(1 - r) when |r| < 1. If |r| ≥ 1, clearly state the series diverges and has no finite sum.
Context interpretation errors: In word problems, identify whether "first term" means the initial value or starts counting from a different point. Read carefully to extract correct values for a and n.
Exam technique for "Algebra: Sequences and Series"
Show working for formula selection: Write down the formula before substituting values. Examiners award method marks even if arithmetic errors occur, but only if they can see your approach clearly.
Multi-part questions build on each other: Results from part (a) often feed into parts (b) or (c). If you cannot complete an early part, use a sensible assumed value and state "using the answer from (a)" to earn follow-through marks.
Recognize command words: "Find" requires a numerical answer with working. "Show that" demands clear algebraic steps leading to the given result. "Determine" or "calculate" need both method and final answer, usually to 2-3 significant figures unless stated otherwise.
Units and context matter: Include appropriate units in real-world problems (dollars, metres, years). State conclusions clearly: "The 15th payment exceeds $5000" rather than leaving "n = 15" without interpretation.
Quick revision summary
Arithmetic progressions use uₙ = a + (n - 1)d and Sₙ = n/2[2a + (n - 1)d]. Geometric progressions use uₙ = arⁿ⁻¹ and Sₙ = a(1 - rⁿ)/(1 - r). Check sequence type by examining differences (constant = AP) or ratios (constant = GP). Infinite GP sums exist only when |r| < 1, giving S∞ = a/(1 - r). Master sigma notation and standard summation formulas. Always show formula selection, substitute carefully, and interpret answers within problem contexts for maximum marks.