2.3 Sequences, Patterns & Algebra Foundations

2.3 Sequences, Patterns & Algebra Foundations

Recognizing numerical patterns, working with prime numbers, and solving algebraic equations are key components of both the Mathematics and Non-Verbal Reasoning sections of the Kent Test. This section covers arithmetic sequences, position-to-term rules, number theory, algebraic simplification, and equation solving.

Arithmetic Sequences & $n\text{th}$ Term Rules

An arithmetic sequence is a list of numbers with a constant common difference between consecutive terms.

Term-to-Term vs. Position-to-Term Rules

• Term-to-term rule: Describes how to get from one term to the next (e.g., "add 5").
• Position-to-term ($n\text{th}$ term) rule: Provides a direct algebraic formula to calculate the $n\text{th}$ term based on its position number $n$.

Step-by-Step Method to Find the $n\text{th}$ Term ($dn + c$)

1. Calculate the common difference ($d$) between consecutive terms. This forms the coefficient of $n$ ($dn$).
2. Compare the $dn$ table with the actual sequence to find the zero-th term ($c = a_1 - d$).
3. Combine into the general expression $dn + c$.

Worked Example: Find the $n\text{th}$ term formula for the sequence $3, 8, 13, 18, 23, \dots$ and calculate the 50th term.

1. Common Difference: $8 - 3 = 5$, $13 - 8 = 5 \rightarrow d = 5$. The term contains $5n$.
2. Zero-th Term ($c$): First term is $3$. Subtract difference $5$: $3 - 5 = -2$.
3. Formula: $n\text{th}\text{ term} = 5n - 2$.
4. 50th Term Calculation: Substitute $n = 50$: $5(50) - 2 = 250 - 2 = 248$.

Special Non-Linear Sequences

• Square Numbers ($n^2$): $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, \dots$
• Cube Numbers ($n^3$): $1, 8, 27, 64, 125, 216, \dots$
• Triangular Numbers ($\frac{n(n+1)}{2}$): $1, 3, 6, 10, 15, 21, 28, \dots$
• Fibonacci Sequence: Each term is the sum of the preceding two terms ($1, 1, 2, 3, 5, 8, 13, 21, \dots$).

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Reading a Sequence Backwards and Sideways

Not every sequence question asks for the next term. Three variants recur, and each needs the rule in a different form.

Find an earlier term. Given 17, 23, 29, 35 and asked for the term before 17, subtract the common difference: $17 - 6 = \mathbf{11}$. Children who only ever practise "what comes next" freeze on this.

Find a distant term. Given 5, 9, 13, 17 and asked for the 30th term, do not count forwards. Use the position-to-term rule: $d = 4$, the zero-th term is $1$, so the rule is $4n + 1$ and the 30th term is $4(30) + 1 = \mathbf{121}$.

Find the position of a value. Given the same sequence, ask whether 83 appears. Solve $4n + 1 = 83$, giving $n = 20.5$. Because $n$ is not a whole number, 83 is not in the sequence. A non-integer answer is the proof, and questions are written to test exactly that reasoning.

Descending and Fractional Sequences

• Descending: 47, 41, 35, 29 has $d = -6$, so the rule is $-6n + 53$. The sign of $d$ must be carried through every step.
• Fractional: $\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}$ increases by $\frac{1}{4}$. Convert to quarters first — $\frac{2}{4}, \frac{3}{4}, \frac{4}{4}, \frac{5}{4}$ — and the pattern becomes obvious.
• Decimal: 0.3, 0.55, 0.8, 1.05 increases by 0.25. Reading the gap once, precisely, prevents a run of wrong terms.

Prime numbers, factors, HCF and LCM support the harder sequence and pattern questions, and they are covered in full in section 2.6.

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Algebraic Notation, Simplification & Substitution

Algebra uses letters (variables) to represent unknown values.

Key Rules of Algebraic Notation

• $3 \times x$ is written as $3x$.
• $x \div 4$ is written as $\frac{x}{4}$.
• $a \times b$ is written as $ab$.
• $x \times x$ is written as $x^2$.

Collecting Like Terms

Only terms with exact matching variable combinations can be combined.

• Example: Simplify $4x + 3y - 2x + 7y - 5$.
  • Combine $x$-terms: $4x - 2x = 2x$.
  • Combine $y$-terms: $3y + 7y = 10y$.
  • Constant: $-5$.
  • Simplified expression: $2x + 10y - 5$.

Algebraic Substitution

Replace variables with their given numerical values and calculate using BIDMAS.

Worked Example: Evaluate $2a^2 - 3bc$ when $a = 3$, $b = -2$, and $c = 5$.

1. Evaluate $a^2$: $3^2 = 9$.
2. First term: $2 \times 9 = 18$.
3. Second term: $3 \times (-2) \times 5 = 3 \times (-10) = -30$.
4. Subtract: $18 - (-30) = 18 + 30 = 48$.

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Solving Linear Equations & Function Machines

To solve an equation, perform inverse operations equally to both sides to isolate the unknown variable.

Inverse Operation Pairs

• Addition ($\text{+}) \leftrightarrow$ Subtraction ($\text{-}$)
• Multiplication ($\times) \leftrightarrow$ Division ($\div$)
• Squaring ($x^2) \leftrightarrow$ Square Root ($\sqrt{x}$)

Solving Equations with Unknowns on Both Sides

Worked Example: Solve $7x - 5 = 3x + 11$.

1. Subtract $3x$ from both sides: $7x - 3x - 5 = 11 \rightarrow 4x - 5 = 11$.
2. Add $5$ to both sides: $4x = 11 + 5 \rightarrow 4x = 16$.
3. Divide both sides by $4$: $x = \frac{16}{4} \rightarrow x = 4$.

Function Machines

A function machine applies a sequence of operations to an input number to yield an output. To find an unknown input when given the output, work backwards from right to left using inverse operations.

• Example: $\text{Input} \rightarrow [\times 4] \rightarrow [-7] \rightarrow \text{Output } 29$.
• Reverse: $29 \xrightarrow{+7} 36 \xrightarrow{\div 4} 9$. Input = $9$.