Algebra and Sequences

Algebra might sound complicated, but it is simply using letters to represent unknown numbers. In the FSCE 11+ exam, you will need to solve simple equations, substitute values into expressions, work with function machines, and find patterns in number sequences. This lesson breaks down each skill with clear explanations and worked examples.

What Is Algebra?

In algebra, we use letters (usually x, y, n, or a) to stand for numbers we do not know yet. For example:

• x + 5 = 12 means "some number plus 5 equals 12." You can work out that x = 7.
• 3n means "3 times n." If n = 4, then 3n = 12.

Important Conventions

• 3n means 3 x n (we do not write the multiplication sign).
• n² means n x n (n squared).
• 2n + 1 means "double n, then add 1."

Solving Simple Equations

An equation is like a balance — whatever you do to one side, you must do to the other.

One-Step Equations

• x + 7 = 15 → Subtract 7 from both sides → x = 8
• x - 3 = 11 → Add 3 to both sides → x = 14
• 4x = 28 → Divide both sides by 4 → x = 7
• x/5 = 6 → Multiply both sides by 5 → x = 30

Two-Step Equations

• 3x + 4 = 19 → Subtract 4 from both sides → 3x = 15 → Divide by 3 → x = 5
• 2x - 7 = 9 → Add 7 to both sides → 2x = 16 → Divide by 2 → x = 8

Golden rule: Always undo the operations in reverse order. If the expression says "multiply by 3, then add 4," undo by subtracting 4 first, then dividing by 3.

Substitution

Substitution means replacing a letter with a number and calculating the result.

If a = 3 and b = 5:

• a + b = 3 + 5 = 8
• 2a + 3b = 2(3) + 3(5) = 6 + 15 = 21
• ab = 3 x 5 = 15
• a² + b = 9 + 5 = 14
• 4(a + b) = 4(3 + 5) = 4 x 8 = 32

Function Machines

A function machine takes an input number, performs one or more operations, and produces an output.

Example: Input → x3 → +2 → Output

If the input is 5: 5 → x3 = 15 → +2 = 17. Output is 17.

To work backwards (finding the input from the output), reverse the operations: If the output is 17: 17 → -2 = 15 → ÷3 = 5. Input is 5.

flowchart LR
    A[Input] --> B[x 3]
    B --> C[+ 2]
    C --> D[Output]
    D -.->|Reverse: - 2| C
    C -.->|Reverse: ÷ 3| B

Number Sequences

A sequence is a list of numbers that follows a rule. You need to recognise different types and find missing terms.

Linear Sequences

Each term increases (or decreases) by the same amount. This amount is called the common difference.

• 3, 7, 11, 15, 19, ... → common difference is +4.
• 20, 17, 14, 11, 8, ... → common difference is -3.

Finding the nth Term of a Linear Sequence

The nth term rule lets you find any term without listing them all.

For the sequence 3, 7, 11, 15, 19:

1. Common difference = 4, so the rule starts with 4n.
2. When n = 1: 4(1) = 4, but the first term is 3. So we need to subtract 1.
3. nth term = 4n - 1.
4. Check: n = 2 → 4(2) - 1 = 7. n = 3 → 4(3) - 1 = 11. Correct!

Square Numbers

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...

These are 1², 2², 3², 4², 5², ... The nth term is n².

Triangular Numbers

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...

Each term adds one more than the previous gap: +2, +3, +4, +5, +6, ...

The nth triangular number = n(n+1)/2.

Fibonacci-Type Sequences

Each term is the sum of the two terms before it.

1, 1, 2, 3, 5, 8, 13, 21, 34, ...

Rule: add the last two numbers to get the next. 21 + 34 = 55.

Worked Examples

Worked Example 1: Solving a Two-Step Equation

Question: Solve 5x - 3 = 22

Step-by-step solution:

1. Add 3 to both sides: 5x = 25
2. Divide both sides by 5: x = 5
3. Check: 5(5) - 3 = 25 - 3 = 22. Correct!

Answer: x = 5

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Worked Example 2: Substitution

Question: If p = 4 and q = -2, find the value of 3p² - 2q.