Master arithmetic and geometric sequences from multiple perspectives: identify the type, find the nth term, calculate sums, and prove formulas. At Year 4 Advanced, you must prove why formulas work, not just apply them.
What You'll Learn
Distinguish arithmetic from geometric sequences using constant difference vs constant ratio
Find the nth term of arithmetic and geometric sequences
Calculate the sum of the first n terms of both types
Use sigma notation to express series
Prove the arithmetic series sum formula by pairing
Apply sequences to real-world problems (savings, population growth, depreciation)
IB Assessment Focus
Criterion A: Apply sequence formulas to unfamiliar, multi-step problems (compound interest, population modelling).
Criterion B: Prove the arithmetic series formula by writing the sum forwards and backwards.
Criterion C: Communicate solutions clearly; move between term-by-term, formula, and sigma representations.
Criterion D: Justify degree of accuracy and explain whether solutions are reasonable in context.
Key Vocabulary
Term
Definition
Arithmetic sequence
A sequence with a constant common difference d (e.g., 3, 7, 11, 15… d = 4)
Geometric sequence
A sequence with a constant common ratio r (e.g., 2, 6, 18, 54… r = 3)
Common difference (d)
The fixed amount added to each term in an arithmetic sequence: d = u&sub2; − u&sub1;
Common ratio (r)
The fixed number each term is multiplied by in a geometric sequence: r = u&sub2; / u&sub1;
nth term (un)
A formula giving the value of any term in the sequence by its position
Series
The sum of the terms of a sequence
Sigma notation (Σ)
Compact notation used to express the sum of a series: Σ from i = 1 to n
Convergent series
A geometric series where |r| < 1, so the sum approaches a finite limit as n → ∞
Mathematics · Topic 1.2
Arithmetic Sequences
An arithmetic sequence increases (or decreases) by the same amount each time. The constant difference d determines every term.
Arithmetic: equal steps (+d) each term · Geometric: multiplied by ratio (×r) each term
nth Term Formula
Arithmetic nth Term
un = u1 + (n − 1)d
Example: Find the 25th term of 5, 8, 11, 14, …
Worked Example — nth Term
u1 = 5, d = 8 − 5 = 3identify u₁ and d
u25 = 5 + (25 − 1)(3) = 5 + 72substitute n = 25
u25 = 77
Sum of n Terms
Arithmetic Series Sum
Sn = n2 × (u1 + un) or Sn = n2 × [2u1 + (n − 1)d]
Example: Find the sum of the first 20 terms of 4, 11, 18, 25, …
Worked Example — Arithmetic Series Sum
u1 = 4, d = 7, n = 20identify values
u20 = 4 + 19(7) = 137find last term
S20 = 202 × (4 + 137) = 10 × 141apply sum formula
S20 = 1410
Finding d or u1 from Given Information
Example: The 3rd term of an arithmetic sequence is 14 and the 7th term is 30. Find d and u1.
Common Mistake: Using n instead of (n − 1) in the formula. The first term adds zero differences: u1 = u1 + (1 − 1)d = u1 + 0. Always check your formula gives back u1 when n = 1.
Mathematics · Topic 1.2
Geometric Sequences
A geometric sequence multiplies by the same ratio each time. Geometric sequences grow (or decay) exponentially, making them essential for modelling real-world phenomena.
nth Term Formula
Geometric nth Term
un = u1 × rn−1
Example: Find the 6th term of 3, 12, 48, 192, …
Worked Example — Geometric nth Term
u1 = 3, r = 123 = 4identify u₁ and r
u6 = 3 × 45 = 3 × 1024substitute n = 6
u6 = 3072
Sum of n Terms
Geometric Series Sum (r ≠ 1)
Sn = u1(1 − rn)1 − r
Example: Find the sum of the first 8 terms of 2, 6, 18, 54, …
Worked Example — Geometric Series Sum
u1 = 2, r = 3, n = 8identify values
S8 = 2(1 − 38)1 − 3 = 2(1 − 6561)−2apply sum formula
= 2(−6560)−2simplify numerator
S8 = 6560
Growth vs Decay
Condition
Behaviour
Example
r > 1
Exponential growth
Population doubling: r = 2
0 < r < 1
Exponential decay
Depreciation at 20%: r = 0.8
r < 0
Alternating sign
1, −2, 4, −8… r = −2
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Critical Rule: Always verify whether a sequence is arithmetic or geometric BEFORE applying any formula. Arithmetic: subtract consecutive terms and check for constant difference. Geometric: divide consecutive terms and check for constant ratio.
Mathematics · Topic 1.2
Series & Sigma Notation
A series is the sum of the terms of a sequence. Sigma notation provides a compact way to write sums.
Sigma Notation
The sum of the first 5 terms of 2, 5, 8, 11, 14 can be written as:
Σi=15 (3i − 1) = 2 + 5 + 8 + 11 + 14 = 40
Proof: Arithmetic Series Sum Formula
Prove that Sn = n/2 × (u1 + un).
Write the sum forwards: S = u1 + (u1 + d) + (u1 + 2d) + … + un.
Write the sum backwards: S = un + (un − d) + (un − 2d) + … + u1.
Add both equations term by term: 2S = (u1 + un) + (u1 + un) + … (n times).
This proof is a key Criterion B target at Year 4 — you must be able to reproduce it.
Mathematics · Topic 1.2
Applications of Sequences
Sequences model many real-world situations. Arithmetic sequences model constant change; geometric sequences model percentage-based change.
Common Applications
Situation
Type
Model
Regular savings (fixed deposit)
Arithmetic
Add the same amount each period
Simple interest
Arithmetic
Interest = P × r × t
Compound interest
Geometric
A = P(1 + r)n
Population growth
Geometric
Pn = P0 × rn
Depreciation
Geometric
V = V0(1 − rate)n
Example: A car worth $25,000 depreciates by 15% per year. Find its value after 5 years.
Worked Example — Depreciation (Geometric)
u1 = 25000, r = 1 − 0.15 = 0.85identify model
V = 25000 × 0.855 = 25000 × 0.4437substitute n = 5
V ≈ $11,093 (nearest dollar)
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Criterion D Tip: When solving real-world sequence problems, always justify your rounding. Money is rounded to 2 d.p. (or nearest dollar); populations to whole numbers. State why your degree of accuracy is appropriate for the context.
Mathematics · Topic 1.2
Worked Examples
Multi-step solutions showing the reasoning expected at Year 4 Advanced.
EXAMPLE 1Find the 30th term and the sum of the first 30 terms of 7, 12, 17, 22, …
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Full Solution
Arithmetic sequence: d = 12 − 7 = 5, u1 = 7.
u30 = 7 + (30 − 1)(5) = 7 + 145 = 152.
S30 = 30/2 × (7 + 152) = 15 × 159 = 2385.
Verify: u1 = 7 + 0(5) = 7 ✓, u2 = 7 + 1(5) = 12 ✓
EXAMPLE 2A geometric sequence has u3 = 24 and u6 = 192. Find u1 and r.
APPLYA bacterial colony doubles every hour starting from 500 bacteria. How many bacteria after 12 hours?
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Model Answer
Geometric: u1 = 500, r = 2.
After 12 hours: u13 = 500 × 212 = 500 × 4096 = 2,048,000 bacteria.
(Note: the initial count is u1, so after 12 doublings we need the 13th term, or equivalently 500 × 212.)
PROVEShow that the sum of the first n odd numbers equals n².
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Model Answer
The odd numbers 1, 3, 5, … form an arithmetic sequence with u1 = 1, d = 2.
un = 1 + (n − 1)(2) = 2n − 1.
Sn = n/2 × (u1 + un) = n/2 × (1 + 2n − 1) = n/2 × 2n = n². ∎
Check: n = 4: 1 + 3 + 5 + 7 = 16 = 4² ✓
ANALYSEIs the sequence 2, 6, 18, 54 arithmetic, geometric, or neither? Justify.
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Model Answer
Check arithmetic: 6 − 2 = 4, 18 − 6 = 12. Differences are not equal, so NOT arithmetic.
Check geometric: 6/2 = 3, 18/6 = 3, 54/18 = 3. Ratios are constant (r = 3), so this is geometric.
REAL-WORLDA machine loses 12% of its value each year. If it costs $80,000 new, when is it worth less than $30,000?
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Model Answer
Geometric decay: V = 80000 × 0.88n. Solve 80000 × 0.88n < 30000.
0.88n < 0.375. Taking logarithms: n × log(0.88) < log(0.375).
n > log(0.375)/log(0.88) = (−0.4260)/(−0.0555) ≈ 7.67.
Since n must be a whole number, the machine is worth less than $30,000 after 8 years. Verify: After 7 years: 80000 × 0.887 ≈ $31,648. After 8 years: 80000 × 0.888 ≈ $27,850 < $30,000 ✓
Mathematics · Review
Flashcard Review
Tap each card to reveal the answer. Try to answer from memory first.
State the nth term of an arithmetic sequence.
un = u1 + (n − 1)d where u1 is the first term and d is the common difference.
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State the nth term of a geometric sequence.
un = u1 × rn−1 where u1 is the first term and r is the common ratio.
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How do you find the common difference?
d = u2 − u1 (subtract consecutive terms). If d is constant, the sequence is arithmetic.
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How do you find the common ratio?
r = u2 / u1 (divide consecutive terms). If r is constant, the sequence is geometric.
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Sum of n terms (arithmetic)?
Sn = n/2 × (u1 + un) or Sn = n/2 × [2u1 + (n−1)d]
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Sum of n terms (geometric)?
Sn = u1(1 − rn) / (1 − r) for r ≠ 1.
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What is the key step to prove the arithmetic sum formula?
Write S forwards and backwards, then add. Each pair sums to (u1 + un), and there are n pairs. So 2S = n(u1 + un).
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Compound interest formula?
A = P(1 + r)n — a geometric model where P is principal, r is rate per period, n is number of periods.
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What does r = 0.85 mean in a depreciation context?
The item retains 85% of its value each period (loses 15%). After n periods: V = V0 × 0.85n.
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Sum of first n positive integers?
n(n + 1)/2. This follows directly from the arithmetic sum formula with u1 = 1, un = n.
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When does a geometric series converge?
When |r| < 1. The sum to infinity is S∞ = u1 / (1 − r).
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What is sigma notation?
Σ notation is a compact way to write sums. Σi=1n f(i) means "sum f(i) for i = 1, 2, 3, …, n."
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How do you determine if a term belongs to a sequence?
Set un = the given value and solve for n. If n is a positive integer, the value IS a term in the sequence.
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What is the difference between a sequence and a series?
A sequence is an ordered list of terms. A series is the SUM of those terms.
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Common mistake with the nth term formula?
Using n instead of (n − 1). Remember: u1 = u1 + (1−1)d = u1 + 0. The exponent/multiple is always one less than the position.