IB Diploma Mathematics: Analysis & Approaches SL · Number & Algebra

Mini-Lesson

Number & Algebra

This mini-lesson covers the SL Number & Algebra toolkit: arithmetic and geometric sequences and series (with Σ notation), the laws of exponents and logarithms, the binomial theorem, and standard form.

Work through each screen, answer the questions — some are wordy, some are calculations — and collect ⭐ stars. Press Start when ready.

Sequences

Arithmetic & geometric sequences

A sequence is an ordered list of terms u₁, u₂, u₃, …

An arithmetic sequence adds a fixed common difference d: u_n = u₁ + (n−1)d.
A geometric sequence multiplies by a fixed common ratio r: u_n = u₁ rⁿ⁻¹.

u_n = u₁ + (n−1)d · u_n = u₁·rⁿ⁻¹test which by checking differences (arithmetic) vs ratios (geometric)

Quick check

Name that sequence

?The sequence 2, 6, 18, 54, … continues by the same rule. What kind of sequence is it, and what is the key constant?

Series

Summing a series

A series is the sum of the terms of a sequence.

Sₙ = n/2 (2u₁ + (n−1)d)arithmetic sum — also n/2 (u₁ + uₙ)

Sₙ = u₁(rⁿ − 1)/(r − 1)geometric sum, r ≠ 1

Σ notation: Σ_k=1ⁿ u_k means add u_k from k = 1 to n.

Calculate

Arithmetic nth term

✎An arithmetic sequence has first term u₁ = 3 and common difference d = 5. Find the 10th term u₁₀.

Hint: uₙ = u₁ + (n−1)d = 3 + 9×5.

Convergence & exponents

Sum to infinity and index laws

A geometric series converges only when |r| < 1; then it has a finite sum to infinity:

S_∞ = u₁ / (1 − r) (|r| < 1)

The laws of exponents hold for all real indices:

aᵐ · aⁿ = aᵐ⁺ⁿ
aᵐ / aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ
a⁻ⁿ = 1/aⁿ, a^1/n = ⁿ√a

Quick check

When does it converge?

?A geometric series has first term u₁ = 10. For which common ratio does the sum to infinity S_∞ exist (converge)?

Calculate

Arithmetic sum

✎For the same sequence (u₁ = 3, d = 5), find the sum of the first 20 terms, S₂₀.

Hint: Sₙ = n/2 (2u₁ + (n−1)d) = 10 × (6 + 95).

Logarithms

Logarithms and their laws

A logarithm is the inverse of exponentiation: log_a x = y ⇔ aʸ = x (a > 0, a ≠ 1).

logₐ(xy) = logₐx + logₐyand logₐ(x/y) = logₐx − logₐy, logₐ(xⁿ) = n logₐx

Special values: logₐ1 = 0, logₐa = 1. Natural log ln = log_e, with e ≈ 2.71828.

Sort it

Arithmetic, geometric or neither?

Tap a sequence or rule, then the family it belongs to.

➕ Arithmetic

✖️ Geometric

❔ Neither

Binomial theorem

The binomial theorem

The expansion of (a + b)ⁿ for positive integer n uses binomial coefficients ₙCᵣ:

(a+b)ⁿ = Σ_r=0ⁿ ₙCᵣ aⁿ⁻ʳ bʳₙCᵣ = n! / (r!(n−r)!) — row n of Pascal's triangle

The general term

The term in bʳ is ₙCᵣ aⁿ⁻ʳ bʳ. To find a coefficient, pick the r that gives the required power.

Calculate

Geometric nth term

✎A geometric sequence has u₁ = 5 and common ratio r = 3. Find the 6th term u₆.

Hint: uₙ = u₁ r^(n−1) = 5 × 3⁵ = 5 × 243.

Quick check

Log law

?Written as a single number, log₁₀ 20 + log₁₀ 5 equals:

Applications

Compound interest & growth

Geometric sequences model compound interest and exponential growth. After n periods at rate i per period:

FV = PV(1 + i)ⁿeach period multiplies by the fixed ratio (1 + i) — a geometric sequence

Watch the period: for k compoundings per year over t years at annual rate r, use FV = PV(1 + r/k)^kt.

Calculate

Sum to infinity

✎A geometric series has u₁ = 8 and r = 0.5. Find its sum to infinity S_∞.

Hint: S∞ = u₁/(1 − r) = 8 / (1 − 0.5).

Match it

Match each expression to its value

Tap a statement on the left, then its matching partner on the right.

Expression

Value

Standard form & surds

Standard form and surds

Standard form writes a number as a × 10^k with 1 ≤ |a| < 10 and integer k — essential for very large or small quantities.

Surds are exact irrational roots. Simplify by extracting square factors and rationalise denominators:

√50 = 5√2 · 1/√2 = √2/2

Calculate

Binomial coefficient

✎In the expansion of (x + 2)⁵, find the coefficient of the x² term.

Hint: term in x² is ₅C₃ x² 2³ = 10 × 8.

Quick check

Index law

?Simplify (x³)⁴ ÷ x⁵ to a single power of x.

Strategy

Putting it together

Decide the structure first: is the pattern arithmetic (constant difference), geometric (constant ratio) or neither? Then choose the matching formula.

Exam habit: for "sum to infinity" always check |r| < 1 before applying the formula, and keep exact values (surds, logs) until the final line.

Recap

The big ideas to know

Arithmetic: uₙ = u₁ + (n−1)d, Sₙ = n/2(2u₁ + (n−1)d)

Geometric: uₙ = u₁rⁿ⁻¹, Sₙ = u₁(rⁿ−1)/(r−1), S∞ = u₁/(1−r) for |r|<1

Exponents: aᵐaⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ

Logs: logₐ(xy) = logₐx + logₐy, logₐxⁿ = n logₐx

Binomial: (a+b)ⁿ = Σ ₙCᵣ aⁿ⁻ʳbʳ, ₙCᵣ = n!/(r!(n−r)!)

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