Variable
A symbol whose value can change.
x, y, count, index
📐 Algebra for Programmers, AP & GP
Use variables and equations to model program behaviour, then master arithmetic and geometric progressions for indexed values, loop totals and growth patterns.
🎯 1. Learning Objectives
After completing this topic, you should be able to:
- Identify variables, constants, coefficients, expressions and equations.
- Simplify expressions using the correct order of operations.
- Solve linear equations, paired equations and basic inequalities.
- Use exponents and roots in programming calculations.
- Find any term and the sum of terms in an AP or GP.
- Translate formulas into loop counts, array indices and growth models.
🧩 2. Algebra Building Blocks
Constant
A fixed value in an expression or program.
7 in 3x + 7
Coefficient
A number multiplying a variable.
3 is the coefficient of x in 3x
Equation
A statement that two expressions are equal.
3x + 7 = 25
🧮 3. Simplifying Expressions
Combine only like terms, and always respect operator precedence.
4x + 3 + 2x − 5 = 6x − 2
1. Parentheses
2. Powers
3. × and ÷
4. + and −
Programming connection: The expression
2 + 3 * 4 equals 14, while (2 + 3) * 4 equals 20.
⚖️ 4. Linear Equations
Apply the same inverse operation to both sides until the variable is isolated.
General Form
ax + b = c, a ≠ 0
x = (c − b) / a
Example: 3x + 5 = 20
Subtract 5: 3x = 15
Divide by 3: x = 5
Verify: Substitute the answer back into the original equation.
Here, 3(5) + 5 = 20, so the solution is correct.
🔗 5. Two-Variable Equations and Inequalities
Simultaneous Equations
Two independent equations can determine two unknowns.
x + y = 10
x − y = 2
x = 6, y = 4
x − y = 2
x = 6, y = 4
Inequalities
Solve like an equation, but reverse the sign when multiplying or dividing by a negative number.
−2x > 8 ⇒ x < −4
⚡ 6. Exponents and Roots
| Rule | Meaning | Example |
|---|---|---|
| am × an = am+n | Add exponents | 2³ × 2⁴ = 2⁷ = 128 |
| am ÷ an = am−n | Subtract exponents | 5⁶ ÷ 5² = 5⁴ |
| (am)n = amn | Multiply exponents | (3²)³ = 3⁶ |
| a⁰ = 1 | Zero power | 12⁰ = 1 |
| √a = a1/2 | Square root | √81 = 9 |
💻 7. Algebra in Programming
Array Index Formula
The kth positive even number is calculated directly.
value = 2k
Loop Total
A loop running 1, 2, ..., n times performs:
n(n + 1) / 2 operations
Coordinate Mapping
A row and column can be mapped into one linear index.
index = row × columns + column
Growth Model
Repeated multiplication creates exponential growth.
value = initial × ratiosteps
➕ 8. Arithmetic Progression (AP)
An AP is a sequence in which the difference between consecutive terms is constant.
nth Term
an = a + (n − 1)d
For 4, 7, 10, ..., a = 4 and d = 3.
Sum of First n Terms
Sn = n/2 [2a + (n − 1)d]
Equivalent form: Sn = n/2(a + l), where l is the last term.
✖️ 9. Geometric Progression (GP)
A GP is a sequence in which each term is obtained by multiplying by a constant ratio.
nth Term
an = arn−1
For 2, 6, 18, ..., a = 2 and r = 3.
Finite Sum
Sn = a(rn − 1) / (r − 1), r ≠ 1
If r = 1, every term equals a and Sn = na.
🧪 10. Algebra–AP–GP Explorer
Choose a model, enter its values and click Analyze. Changing an input does not reveal a result until you request it.
Result
🔍 11. AP and GP Compared
Feature
AP
GP
Change
Add d
Multiply by r
nth term
a + (n − 1)d
arn−1
Example
4, 7, 10, 13
2, 6, 18, 54
Coding use
Linear growth and fixed increments
Doubling, recursion and exponential growth
🧾 12. Solved Problems
Problem 1: Solve an Equation
Problem: Solve 5x − 8 = 27.
Solution:
5x = 27 + 8 = 35
x = 35 ÷ 5 = 7
5x = 27 + 8 = 35
x = 35 ÷ 5 = 7
Problem 2: AP Term and Sum
Problem: For 3, 7, 11, ..., find a10 and S10.
Solution:
a10 = 3 + 9(4) = 39
S10 = 10/2(3 + 39) = 210
a10 = 3 + 9(4) = 39
S10 = 10/2(3 + 39) = 210
Problem 3: GP Sum
Problem: Find the sum 2 + 6 + 18 + 54.
Solution:
a = 2, r = 3, n = 4
S4 = 2(3⁴ − 1)/(3 − 1) = 80
a = 2, r = 3, n = 4
S4 = 2(3⁴ − 1)/(3 − 1) = 80
Problem 4: Programming Formula
Problem: A nested loop runs 1 + 2 + ... + 100 times.
Solution:
S = 100(101)/2 = 5,050 executions.
S = 100(101)/2 = 5,050 executions.
✍️ 13. Practice Problems
Solve each problem first, then use Show Solution to verify your method.
1. Simplify: 7x + 4 − 2x + 3.
Combine like terms: 7x − 2x = 5x and 4 + 3 = 7. Answer: 5x + 7.
2. Solve: 4x + 9 = 37.
4x = 28, so x = 7. Verification: 4(7) + 9 = 37.
3. Find the 12th term of the AP 5, 8, 11, ...
a = 5, d = 3. a12 = 5 + 11(3) = 38.
4. Find the 7th term of the GP 2, 4, 8, ...
a = 2, r = 2. a7 = 2 × 2⁶ = 128.
5. A loop processes 5 more records in every round, beginning with 8 records. How many records are processed in round 10?
This is an AP with a = 8, d = 5. a10 = 8 + 9(5) = 53 records.
📝 14. Quick Revision
- Combine only like terms when simplifying an expression.
- Perform the same operation on both sides of an equation.
- Reverse an inequality sign when multiplying or dividing by a negative number.
- AP: constant difference; an = a + (n − 1)d.
- GP: constant ratio; an = arn−1.
- Use a sum formula instead of a loop when only the total is required.
- Substitute an answer into the original condition to verify it.
INTERACTIVE PRACTICE
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📐 Algebra, AP & GP Practice
Complete 20 questions from easy to hard. Your score, every option, correct answers and explanations appear after final submission.