Series Lesson
Hello. I am going to teach you everything you need to know about sequences and series. I will assume you know nothing — and by the end, you will be summing terms, expanding binomials, and solving real-world problems with confidence. Let us begin.
What is a Sequence?
A sequence is an ordered list of numbers that follow a rule. Each number in the sequence is called a term. We write the n-th term as uₙ.
For example: 2, 4, 6, 8, 10, ... is a sequence. The first term u₁ = 2, the second term u₂ = 4, and the n-th term uₙ = 2n.
A series is what you get when you add up the terms of a sequence. So 2 + 4 + 6 + 8 + 10 is a series (it is the sum of the first 5 terms).
Key Vocabulary
🔵 Term — a single number in the sequence (e.g. the 3rd term is 6)
🔵 uₙ — the n-th term formula (tells you the value of the term at position n)
🔵 Common difference — the gap between consecutive terms in an arithmetic sequence
🔵 Common ratio — the multiplier between consecutive terms in a geometric sequence
Look at the pattern: each term increases by 4. This is called the common difference, d = 4.
The sequence goes: 3, 7, 11, 15, 19, ...
u₁ = 3, u₂ = 7 = 3 + 4, u₃ = 11 = 3 + 2×4, u₄ = 15 = 3 + 3×4
So u₅ = 3 + 4×4 = 3 + 16 = 19
Teacher's Tip: A sequence is just a list. A series is a sum. Remember: sequence = list, series = sum. The letter S stands for Sum!
Arithmetic Progressions (AP)
An arithmetic progression (AP) is a sequence where each term increases (or decreases) by the same fixed amount. That fixed amount is called the common difference, written as d.
Here, a = first term (u₁), d = common difference, and n = term number.
Here a = 5, d = 3, n = 10.
u₁₀ = a + (10 − 1)d = 5 + 9×3
= 5 + 27 = 32
Sum of an Arithmetic Series
The sum of the first n terms of an AP is given by two equivalent formulas:
Where l = uₙ = a + (n − 1)d is the last term. Choose whichever is more convenient.
a = 2, d = 3, n = 12.
Using Sₙ = n/2 (2a + (n − 1)d):
S₁₂ = 12/2 (2×2 + 11×3) = 6 (4 + 33)
= 6 × 37 = 222
Teacher's Tip: Think of AP sums like finding the average of the first and last term, then multiplying by the number of terms. That is exactly what n/2 (a + l) does — the average of a and l is (a + l)/2, times n gives the sum.
Geometric Progressions (GP)
A geometric progression (GP) is a sequence where each term is multiplied by the same fixed amount. That fixed multiplier is called the common ratio, written as r.
Here, a = first term, r = common ratio, n = term number.
Here a = 3, r = 2, n = 6.
u₆ = 3 × 2⁶⁻¹ = 3 × 2⁵
= 3 × 32 = 96
Sum of a Geometric Series
The sum of the first n terms of a GP:
Infinite Geometric Series
If |r| < 1, the terms get smaller and smaller, and the sum converges to a finite value:
a = 2, r = 3, n = 8.
S₈ = 2(1 − 3⁸) / (1 − 3)
= 2(1 − 6561) / (−2)
= 2(−6560) / (−2) = 6560
a = 10, r = 0.5. Since |0.5| < 1, we can use S∞.
S∞ = a / (1 − r) = 10 / (1 − 0.5)
= 10 / 0.5 = 20
Teacher's Tip: For a GP, always check r first. If |r| ≥ 1, the infinite sum does not exist (it diverges). If |r| < 1, the terms get smaller and smaller, and the sum approaches a finite value — that is the sum to infinity.
Sigma Notation
Sigma notation (Σ) is a compact way to write a series. The Greek letter Σ (sigma) means "sum".
The variable below Σ (i = 1) is the starting index. The number above Σ (n) is the ending index. The expression to the right tells you what to sum.
Substitute i = 1, 2, 3, 4, 5:
= (2×1 + 1) + (2×2 + 1) + (2×3 + 1) + (2×4 + 1) + (2×5 + 1)
= 3 + 5 + 7 + 9 + 11
= 35
Sigma Notation Explorer
Use the sliders below to adjust the limits and see the sum update in real time.
Sum = 3 + 5 + 7 + 9 + 11 = 35
Properties of Sigma Notation
🔵 Constant multiple: Σ c·f(i) = c · Σ f(i)
🔵 Sum of constants: Σ c = c × n
🔵 Splitting: Σ (f(i) + g(i)) = Σ f(i) + Σ g(i)
Teacher's Tip: Sigma notation is just a shorthand. If you ever feel confused, write out the first few terms explicitly. The pattern will reveal itself.
Binomial Expansion
A binomial is an expression with two terms added together, like (a + b). The binomial expansion tells us how to expand (a + b)ⁿ for any positive integer n.
The symbol ⁿCᵣ (read "n choose r") is called a binomial coefficient. It counts how many ways you can choose r items from n items.
The exclamation mark ! means factorial. For example: 5! = 5 × 4 × 3 × 2 × 1 = 120.
Pascal's Triangle
The binomial coefficients follow a beautiful pattern called Pascal's Triangle. Each number is the sum of the two numbers above it.
From Pascal's triangle, the coefficients for n=4 are: 1, 4, 6, 4, 1
(x + 2)⁴ = 1·x⁴·2⁰ + 4·x³·2¹ + 6·x²·2² + 4·x¹·2³ + 1·x⁰·2⁴
= x⁴ + 8x³ + 24x² + 32x + 16
Interactive Binomial Expander
Enter values for a, b, and n, and see every term of the expansion.
Teacher's Tip: Pascal's triangle is worth memorising up to n=5. For larger n, use the ⁿCᵣ formula. Remember: the powers of a go down (n, n−1, ..., 0) while the powers of b go up (0, 1, ..., n), and they always sum to n.
Applications of Series
Sequences and series are not just abstract mathematics — they appear everywhere in the real world.
Compound Interest
If you invest £P at an annual interest rate of r (as a decimal), after n years the amount is:
This is a geometric progression with common ratio (1 + r). Each year multiplies the previous amount by (1 + r).
P = 1000, r = 0.05, n = 10
A = 1000(1 + 0.05)¹⁰ = 1000(1.05)¹⁰
= 1000 × 1.62889... = £1628.89
Population Growth
If a population grows by a fixed percentage each year, the population after n years is another GP:
Where P₀ is the initial population and g is the growth rate (as a decimal).
P₀ = 5000, g = 0.03, n = 6
P₆ = 5000(1.03)⁶
= 5000 × 1.19405... = 5970
Loan Repayment
When repaying a loan with fixed monthly payments, the outstanding balance follows a pattern that uses a GP sum. The total amount repaid over n months with monthly payment M is simply:
If there is interest, the calculation uses a GP sum formula to find the present value of all payments.
Teacher's Tip: Whenever you see something growing by a constant percentage, think "geometric progression." Whenever you see something changing by a constant amount, think "arithmetic progression." That distinction alone solves most problems.
Practice — Step by Step
Solve each problem one step at a time. Type your answer for each step and click Check Step. You must get each step correct before moving to the next.
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- AP vs GP at a glance: AP adds the same amount each time → uₙ = a + (n−1)d. GP multiplies by the same amount each time → uₙ = arⁿ⁻¹. If you see "add" think AP, if you see "multiply" think GP.
- Sum formulas memory trick: AP sum = (number of terms) × (average of first and last). GP sum = (first term) × (1 − rⁿ) / (1 − r). The GP formula only works when r ≠ 1.
- Binomial coefficient pattern: For (a + b)ⁿ, the powers of a go n, n−1, ..., 0 and powers of b go 0, 1, ..., n. They always add to n. The coefficients come from Pascal's triangle row n.
- Sigma shortcut: Σ i from 1 to n = n(n+1)/2. Σ i² = n(n+1)(2n+1)/6. These save time in exams.
- Infinite sum check: S∞ only exists if |r| < 1. If r ≥ 1, the sum diverges to infinity. If r ≤ −1, it oscillates and diverges.
- uₙ is the n-th term: The formula uₙ = a + (n−1)d (AP) or uₙ = arⁿ⁻¹ (GP) gives the value at position n. Always check whether you need the term or the sum.
- Sₙ is the sum of first n terms: Do not confuse Sₙ with uₙ. Sₙ adds everything from u₁ to uₙ. uₙ is just one term.
- Sigma notation is flexible: The index variable can be anything (i, k, r, etc.). Σ from i=m to n means plug in i=m, m+1, ..., n and add them.
- Binomial theorem works for any positive integer n: The expansion has n+1 terms. The sum of the powers of a and b in each term is always n.
- Check the common ratio: In a GP, divide any term by the previous term to find r. If the ratio is not constant, it is not a GP.
- Confusing AP and GP formulas: uₙ = a + (n−1)d is for AP (addition). uₙ = arⁿ⁻¹ is for GP (multiplication). Mixing them up is the #1 error. Look at the pattern before choosing.
- Using S∞ when |r| ≥ 1: The sum to infinity only exists if |r| < 1. If r = 2, the terms get bigger and the sum goes to infinity — you cannot use a/(1−r).
- Forgetting the rⁿ term in Sₙ: The GP sum formula is a(1−rⁿ)/(1−r), not a(1−r)/(1−r). You must include rⁿ or you will get the wrong answer.
- Sigma index errors: Σᵢ₌₃⁷ f(i) means i = 3, 4, 5, 6, 7 — that is 5 terms, not 7−3=4 terms. Count carefully: (upper − lower + 1) terms.
- Binomial sign mistakes: If b is negative, the signs alternate. (x − 2)⁴ has alternating positive and negative terms. Write (x + (−2))⁴ and use the formula correctly.
The Golden Rule of Series
Identify the pattern first. Is it adding a constant (AP) or multiplying by a constant (GP)? Once you know the type, the formulas follow. When in doubt, write out the first few terms — the pattern always reveals itself.
You Have Completed the Series Lesson
Here is what you have learned:
- ✅ What sequences and series are, term notation uₙ
- ✅ Arithmetic Progressions: uₙ = a + (n−1)d, Sₙ = n/2(2a + (n−1)d)
- ✅ Geometric Progressions: uₙ = arⁿ⁻¹, Sₙ = a(1−rⁿ)/(1−r), S∞ = a/(1−r)
- ✅ Sigma notation Σ and how to evaluate sums
- ✅ Binomial expansion: (a+b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ and Pascal's triangle
- ✅ Real-world applications: compound interest, population growth, loan repayment
Well done. You now know series. From arithmetic patterns to infinite sums — this is the foundation of much of Pure Mathematics.