Number and Algebra Formulae AA SL & AA HL: Complete IB Mathematics Guide
Welcome to the definitive guide for Number and Algebra Formulae essential for IB Mathematics Analysis and Approaches at both Standard Level and Higher Level. This comprehensive resource covers fundamental algebraic sequences including arithmetic and geometric progressions, finite and infinite series summation formulas, compound interest calculations for financial mathematics, exponential and logarithmic functions with their conversion rules and laws, and the binomial theorem with binomial coefficients for polynomial expansion. These core topics form the algebraic foundation for both SL and HL students, appearing frequently in Paper 1 and Paper 2 examinations, providing essential tools for modeling real-world phenomena from population growth to financial investments, and serving as prerequisites for advanced calculus, statistics, and mathematical modeling throughout the IB Mathematics AA curriculum.
Understanding AA SL & AA HL Content Coverage AA SL & HL
The number and algebra formulae presented in this guide apply to both Standard Level and Higher Level students in IB Mathematics Analysis and Approaches. These topics constitute the algebraic core that all AA students must master regardless of level. Both SL and HL students study arithmetic sequences with nth term and sum formulas, geometric sequences including convergent and divergent series, compound interest for financial applications, exponential and logarithmic functions with their interconversion and laws, and the binomial theorem for positive integer indices. The key distinction emerges in depth of application rather than formula coverage: HL students encounter more complex problem types requiring deeper algebraic manipulation, multi-step reasoning combining several topics simultaneously, proof-based questions demanding rigorous mathematical justification, and connections to HL-only calculus topics like Maclaurin series and differential equations. Nevertheless, the fundamental formulas remain identical, making this guide invaluable for both levels with supplementary HL-only topics covered in our dedicated Higher Level resources.
Arithmetic Sequences AA SL & HL
Understanding Arithmetic Sequences
An arithmetic sequence (also called arithmetic progression) is a sequence of numbers where consecutive terms differ by a constant value called the common difference, denoted d. This regular additive pattern makes arithmetic sequences perfect for modeling situations with constant growth or decay: linear savings plans, uniform temperature changes, regular payment schedules, or evenly spaced measurements.
where:
u_n = nth term (the term you want to find)
u_1 = first term of the sequence
n = position/term number
d = common difference (u₂ - u₁)
Key insight: To get from u₁ to u_n, you add d exactly (n-1) times
where:
S_n = sum of the first n terms
Two equivalent forms:
Form 1: Use when you know u₁, n, and d
Form 2: Use when you know first term u₁ and last term u_n
Interpretation: Average of first and last term, multiplied by number of terms
An arithmetic sequence has first term u₁ = 5 and common difference d = 3.
(a) Find the 20th term.
Solution: Use u_n = u₁ + (n-1)d
\( u_{20} = 5 + (20-1)(3) = 5 + 19(3) = 5 + 57 = 62 \)
(b) Find the sum of the first 20 terms.
Solution: Use \( S_n = \frac{n}{2}(u_1 + u_n) \)
\( S_{20} = \frac{20}{2}(5 + 62) = 10(67) = 670 \)
Alternative method: Use \( S_n = \frac{n}{2}(2u_1 + (n-1)d) \)
\( S_{20} = \frac{20}{2}(2(5) + 19(3)) = 10(10 + 57) = 10(67) = 670 \) ✓
To find the common difference d of an arithmetic sequence:
Method 1: Subtract any term from the following term: d = u₂ - u₁ = u₃ - u₂ = uₙ - uₙ₋₁
Method 2: If you know two non-consecutive terms, use: d = (u_m - u_n)/(m - n)
Example: Sequence: 7, 10, 13, 16, ... → d = 10 - 7 = 3
Geometric Sequences AA SL & HL
Understanding Geometric Sequences
A geometric sequence (also called geometric progression) is a sequence where consecutive terms have a constant ratio called the common ratio, denoted r. This multiplicative pattern makes geometric sequences ideal for modeling exponential growth or decay: compound interest, population growth, radioactive decay, viral spreading, and doubling/halving processes.
where:
u_n = nth term
u_1 = first term
r = common ratio (u₂/u₁)
n = term number
Key insight: To get from u₁ to u_n, multiply by r exactly (n-1) times
Power is (n-1): Because u₁ already has r⁰ = 1 factor of r
where:
S_n = sum of first n terms
Two equivalent forms:
Form 1: \( \frac{u_1(r^n - 1)}{r - 1} \) (preferred when r > 1)
Form 2: \( \frac{u_1(1 - r^n)}{1 - r} \) (preferred when 0 < r < 1)
Special case: If r = 1, then S_n = n·u₁ (constant sequence)
where:
S_∞ = sum to infinity
CRITICAL CONDITION: |r| < 1 (i.e., -1 < r < 1)
Convergence:
If |r| < 1: series converges (has finite sum)
If |r| ≥ 1: series diverges (no finite sum)
Why it works: As n → ∞, r^n → 0 when |r| < 1
A geometric sequence has first term u₁ = 2 and common ratio r = 3.
(a) Find the 8th term.
Solution: Use \( u_n = u_1 r^{n-1} \)
\( u_8 = 2 \times 3^{8-1} = 2 \times 3^7 = 2 \times 2187 = 4374 \)
(b) Find the sum of the first 8 terms.
Solution: Use \( S_n = \frac{u_1(r^n - 1)}{r - 1} \) (since r > 1)
\( S_8 = \frac{2(3^8 - 1)}{3 - 1} = \frac{2(6561 - 1)}{2} = \frac{2(6560)}{2} = 6560 \)
Find the sum to infinity of the geometric series: 12 + 6 + 3 + 1.5 + ...
Step 1: Identify u₁ and r
u₁ = 12
\( r = \frac{6}{12} = \frac{1}{2} = 0.5 \)
Step 2: Check convergence condition
\( |r| = |0.5| = 0.5 < 1 \) ✓ Series converges
Step 3: Calculate S_∞
\( S_\infty = \frac{u_1}{1 - r} = \frac{12}{1 - 0.5} = \frac{12}{0.5} = 24 \)
Answer: The sum to infinity is 24
Never use the infinite sum formula S_∞ = u₁/(1-r) unless |r| < 1!
Common mistakes:
- Using the formula when r = 1 (gives division by zero)
- Using the formula when |r| > 1 (series diverges to ±∞)
- Using the formula when r = -1 (series oscillates)
Always check: -1 < r < 1 before applying infinite sum formula
Compound Interest AA SL & HL
Understanding Compound Interest
Compound interest is interest calculated on both the initial principal and the accumulated interest from previous periods. Unlike simple interest (linear growth), compound interest produces exponential growth, making investments grow faster over time through the power of compounding. This fundamental financial concept appears throughout real-world applications: savings accounts, loans, investments, inflation calculations, and population modeling.
where:
FV = Future Value (amount after time period)
PV = Present Value (initial principal/investment)
r% = nominal annual rate of interest (as percentage)
k = number of compounding periods per year
n = number of years
Common compounding periods:
- Annually: k = 1
- Semi-annually: k = 2
- Quarterly: k = 4
- Monthly: k = 12
- Daily: k = 365
Sarah invests $5000 at an annual interest rate of 6% compounded quarterly.
(a) What will the investment be worth after 3 years?
Solution:
PV = 5000, r = 6, k = 4 (quarterly), n = 3
\( FV = 5000 \times \left(1 + \frac{6}{100 \times 4}\right)^{4 \times 3} \)
\( = 5000 \times \left(1 + \frac{6}{400}\right)^{12} \)
\( = 5000 \times (1 + 0.015)^{12} \)
\( = 5000 \times (1.015)^{12} \)
\( = 5000 \times 1.19562 \)
\( = 5978.09 \)
Answer: $5978.09 (or $5978 to nearest dollar)
(b) How much interest was earned?
Interest = FV - PV = 5978.09 - 5000 = $978.09
Compound interest is fundamentally a geometric sequence where:
First term: u₁ = PV
Common ratio: \( r = 1 + \frac{\text{rate}}{100k} \)
nth term: \( u_n = PV \times r^{n-1} \)
Each compounding period multiplies the previous amount by the same ratio, creating exponential growth characteristic of geometric progressions.
Exponentials and Logarithms AA SL & HL
Understanding the Exponential-Logarithm Relationship
Exponentials and logarithms are inverse operations: exponentials raise a base to a power to get a result, while logarithms determine what power to raise a base to in order to get a result. This fundamental inverse relationship allows us to solve exponential equations by converting them to logarithmic form and vice versa, making logarithms essential tools for solving growth, decay, and scaling problems across mathematics and science.
Meaning: "a to what power equals b?" Answer: \( \log_a b \)
Key components:
a = base (must be positive, not equal to 1)
x = exponent/power
b = result (must be positive)
Examples:
2³ = 8 ⟺ log₂(8) = 3
10² = 100 ⟺ log₁₀(100) = 2
Product Law:
\[ \log_a(xy) = \log_a x + \log_a y \]Quotient Law:
\[ \log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y \]Power Law:
\[ \log_a(x^m) = m \log_a x \]Change of Base Formula:
\[ \log_a x = \frac{\log_b x}{\log_b a} \]Memory aids:
Multiplication → Addition
Division → Subtraction
Power → Multiplication (bring power down)
Solve: \( 2^{x+1} = 32 \)
Method 1: Matching bases
Rewrite 32 as a power of 2: 32 = 2⁵
\( 2^{x+1} = 2^5 \)
Since bases are equal: x + 1 = 5
Therefore: x = 4
Method 2: Using logarithms
Take log₂ of both sides: \( \log_2(2^{x+1}) = \log_2(32) \)
Apply power law: \( (x+1) \log_2(2) = \log_2(32) \)
Since log₂(2) = 1: x + 1 = log₂(32) = 5
Therefore: x = 4 ✓
Simplify: \( \log_3(27) + \log_3(9) - \log_3(81) \)
Solution:
Apply product and quotient laws:
\( = \log_3\left(\frac{27 \times 9}{81}\right) \)
\( = \log_3\left(\frac{243}{81}\right) \)
\( = \log_3(3) \)
\( = 1 \)
Alternative: Convert to powers of 3
\( \log_3(27) = \log_3(3^3) = 3 \)
\( \log_3(9) = \log_3(3^2) = 2 \)
\( \log_3(81) = \log_3(3^4) = 4 \)
Therefore: 3 + 2 - 4 = 1 ✓
Common logarithm (base 10):
Written as log(x) or log₁₀(x)
Used in scientific calculations, pH, decibels
Natural logarithm (base e):
Written as ln(x) or log_e(x) where e ≈ 2.71828
Used in calculus, continuous growth, exponential decay
Calculator: Use LOG button for log₁₀, LN button for log_e
Binomial Theorem AA SL & HL
Understanding Binomial Expansion
The binomial theorem provides a systematic method to expand expressions of the form (a + b)ⁿ where n is a positive integer. Rather than multiplying out the brackets repeatedly (tedious and error-prone for large n), the theorem gives us a formula involving binomial coefficients that directly determines each term in the expansion. This powerful algebraic tool appears in probability theory (binomial distribution), approximation methods, series expansions, and polynomial algebra throughout mathematics.
General term (r+1)th term:
\[ T_{r+1} = \binom{n}{r}a^{n-r}b^r \]Key observations:
- Powers of a decrease from n to 0
- Powers of b increase from 0 to n
- Sum of powers in each term = n
- Number of terms = n + 1
- Coefficients follow Pascal's triangle
where:
n! (n factorial) = n × (n-1) × (n-2) × ... × 2 × 1
0! = 1 (by definition)
Calculator: Use nCr button
Alternative notation: \( \binom{n}{r} \) or ⁿCᵣ or C(n,r)
Symmetry property: \( \binom{n}{r} = \binom{n}{n-r} \)
Example: ⁵C₂ = ⁵C₃ = 10
Expand \( (2x + 3)^4 \) using the binomial theorem.
Solution: Here a = 2x, b = 3, n = 4
Using \( (a+b)^n = \sum_{r=0}^{n} \binom{n}{r}a^{n-r}b^r \):
Term 1 (r=0): \( \binom{4}{0}(2x)^4(3)^0 = 1 \times 16x^4 \times 1 = 16x^4 \)
Term 2 (r=1): \( \binom{4}{1}(2x)^3(3)^1 = 4 \times 8x^3 \times 3 = 96x^3 \)
Term 3 (r=2): \( \binom{4}{2}(2x)^2(3)^2 = 6 \times 4x^2 \times 9 = 216x^2 \)
Term 4 (r=3): \( \binom{4}{3}(2x)^1(3)^3 = 4 \times 2x \times 27 = 216x \)
Term 5 (r=4): \( \binom{4}{4}(2x)^0(3)^4 = 1 \times 1 \times 81 = 81 \)
Final expansion:
\( (2x + 3)^4 = 16x^4 + 96x^3 + 216x^2 + 216x + 81 \)
Binomial coefficients form Pascal's triangle, where each number is the sum of the two numbers above it:
n=0: 1
n=1: 1 1
n=2: 1 2 1
n=3: 1 3 3 1
n=4: 1 4 6 4 1
n=5: 1 5 10 10 5 1
n=6: 1 6 15 20 15 6 1
Row n gives coefficients for (a+b)ⁿ expansion
Example: (a+b)³ has coefficients 1, 3, 3, 1 from row n=3
Interactive Sequence Calculator
Arithmetic & Geometric Sequence Calculator
Arithmetic Sequence Calculator
Geometric Sequence Calculator
Comparison: Arithmetic vs Geometric Sequences
| Feature | Arithmetic Sequence | Geometric Sequence |
|---|---|---|
| Pattern | Constant difference (add/subtract) | Constant ratio (multiply/divide) |
| nth Term | \( u_n = u_1 + (n-1)d \) | \( u_n = u_1 r^{n-1} \) |
| Growth Type | Linear | Exponential |
| Sum Formula | \( S_n = \frac{n}{2}(2u_1 + (n-1)d) \) | \( S_n = \frac{u_1(r^n - 1)}{r - 1} \) |
| Infinite Sum | Does not converge (±∞) | Converges if |r| < 1: \( S_\infty = \frac{u_1}{1-r} \) |
| Example | 2, 5, 8, 11, 14, ... (d=3) | 2, 6, 18, 54, 162, ... (r=3) |
| Real-World Use | Regular savings, linear depreciation | Compound interest, population growth |
Common Mistakes to Avoid
| Common Error | Correct Approach | Example |
|---|---|---|
| Using power n instead of (n-1) in sequence formulas | Geometric: uₙ = u₁r^(n-1) NOT u₁rⁿ | 3rd term: u₃ = u₁r² (power is 2, not 3) |
| Applying infinite sum when |r| ≥ 1 | Check |r| < 1 before using S∞ = u₁/(1-r) | If r = 2, series diverges - no finite sum |
| Confusing logarithm laws (especially quotient) | log(x/y) = log x - log y (NOT log x + log y) | log₂(8/4) = log₂8 - log₂4 = 3 - 2 = 1 |
| Forgetting to divide by 100k in compound interest | FV = PV(1 + r/100k)^(kn) where r is percentage | 6% quarterly: rate per period = 6/(100×4) = 0.015 |
| Wrong binomial coefficient calculation | ⁵C₂ = 5!/(2!×3!) = 10 (NOT 5!/2!) | Always divide by both r! and (n-r)! |
| Mixing up term number and term value | n is position, uₙ is value at that position | In 2, 5, 8: n=2 gives u₂=5 (not 2) |
Study Strategies for Success
Mastering Sequences and Series
- Identify sequence type first: Check if terms have constant difference (arithmetic) or constant ratio (geometric)
- Write down what you know: List u₁, d or r, n, and what you're finding before choosing formula
- Check your answer makes sense: Does the term fit the pattern? Is the sum reasonable?
- Practice both formula forms: For Sₙ, know when to use each equivalent form
- Always verify convergence: Before using infinite sum, confirm |r| < 1
Mastering Exponentials and Logarithms
- Understand the inverse relationship: Practice converting between aˣ = b and x = log_a(b)
- Memorize the four logarithm laws: Product, quotient, power, change of base
- Look for opportunities to match bases: Often easier than using logarithms directly
- Know your special logs: log₁₀(10) = 1, log₁₀(100) = 2, ln(e) = 1, ln(e²) = 2
- Check domain restrictions: Can only take logarithm of positive numbers
Mastering Binomial Theorem
- Learn Pascal's triangle: Quick way to find coefficients for small n
- Use calculator for nCr: Don't calculate factorials manually for large numbers
- Track powers carefully: Powers of a decrease, powers of b increase, sum = n
- Practice finding specific terms: Use general term formula \( T_{r+1} = \binom{n}{r}a^{n-r}b^r \)
- Remember the (r+1) offset: The term with bʳ is actually the (r+1)th term
Real-World Applications
Where These Topics Appear
- Finance & Economics: Compound interest, loan repayments, investment growth, inflation modeling
- Population Studies: Geometric sequences model exponential population growth/decay
- Physics & Engineering: Radioactive decay (exponential), projectile motion, signal processing
- Computer Science: Algorithm complexity, binary trees, recursion analysis use geometric series
- Biology: Cell division (geometric), bacterial growth, drug concentration decay
- Probability & Statistics: Binomial distribution uses binomial coefficients
- Chemistry: pH scale uses logarithms, half-life calculations use exponentials
Exam Preparation Checklist
- ✓ Identify arithmetic vs geometric sequences quickly
- ✓ Apply nth term formulas: uₙ = u₁ + (n-1)d and uₙ = u₁r^(n-1)
- ✓ Calculate sums using appropriate formulas (finite and infinite)
- ✓ Check convergence condition |r| < 1 before using infinite sum
- ✓ Solve compound interest problems with correct compounding periods
- ✓ Convert between exponential and logarithmic forms fluently
- ✓ Apply all four logarithm laws correctly
- ✓ Use change of base formula when needed
- ✓ Expand binomial expressions using binomial theorem
- ✓ Calculate binomial coefficients nCr accurately
- ✓ Find specific terms in binomial expansions
- ✓ Solve multi-step problems combining several topics
- ✓ Show clear working and use proper notation
- ✓ Practice past paper questions under timed conditions
RevisionTown Resources
Enhance your number and algebra mastery with these comprehensive RevisionTown resources:
- Functions Formulae AA SL & AA HL - Essential functions for all AA students
- Number and Algebra Formulae AA HL Only - Advanced HL-only algebra topics
- Functions Formulae AA HL Only - Advanced HL functions
- Prior Learning Formulae AA SL & AA HL - Foundation skills
- Calculus Formulae AA SL & AA HL - Apply algebra to calculus
- Calculus Formulae AA HL Only - Advanced HL calculus
- IB Diploma Points Calculator - Track your progress
- Grade Calculator - Monitor performance
- IB Mathematics AA vs AI Guide - Course comparison
Technology and GDC Skills
- Sequence mode: Generate terms of sequences automatically
- Sum function: Calculate Sₙ for arithmetic/geometric series
- nCr function: Calculate binomial coefficients
- Log and Ln buttons: Calculate common and natural logarithms
- Exponential key: Use for compound interest calculations
- Table feature: Visualize sequence patterns
- Solver function: Solve exponential and logarithmic equations
- Store values: Use memory for multi-step calculations
Connecting to Other AA Topics
Number and algebra topics integrate throughout the AA curriculum:
- Calculus: Sequences lead to limits; series relate to integration
- Functions: Exponential and logarithmic functions for modeling
- Statistics: Binomial distribution uses binomial coefficients
- Probability: Geometric probability uses geometric series
- Financial Math: Compound interest, loans, annuities
- Differential Equations: Exponential solutions to ODEs
- Complex Numbers (HL): Geometric sequences in polar form
Conclusion
Mastering number and algebra formulae is fundamental for success in IB Mathematics Analysis and Approaches at both Standard and Higher Levels. The topics covered in this comprehensive guide—arithmetic sequences with linear growth patterns, geometric sequences with exponential behavior, compound interest for financial modeling, exponential and logarithmic functions with their inverse relationship and laws, and the binomial theorem for polynomial expansion—form the algebraic backbone supporting virtually all other topics in the AA curriculum from calculus to statistics to mathematical modeling.
Success in this foundational content requires more than formula memorization—it demands conceptual understanding of when and why each formula applies, ability to identify sequence types and choose appropriate solution methods, fluency in converting between different representations (especially exponential-logarithmic conversion), recognition of real-world contexts where these mathematical structures appear, careful attention to domain restrictions and convergence conditions, and systematic problem-solving approaches combining multiple concepts. Whether you're calculating the sum of an infinite geometric series, solving exponential equations using logarithms, applying compound interest to financial scenarios, or expanding binomial expressions, these fundamental algebraic tools are essential.
Regular practice with IB past papers covering these topics, consistent use of your GDC to verify calculations and explore patterns, systematic review of formula applications and common mistake avoidance, understanding of connections between sequences, functions, and real-world models, and building strong algebraic manipulation skills will ensure comprehensive mastery necessary for examination success and mathematical confidence.
Continue building your AA mathematics expertise through RevisionTown's extensive collection of IB resources, practice with our interactive calculators, and connect these algebraic foundations to advanced topics in calculus, statistics, and mathematical modeling throughout your IB journey. Master these number and algebra formulae completely, develop strong pattern recognition and algebraic reasoning skills, and you'll be thoroughly prepared for IB examinations and the quantitative challenges that await in university mathematics, engineering, economics, sciences, and any field requiring sophisticated mathematical thinking. These fundamental algebraic tools are your foundation—build it strong!
