Higher Maths
Recurrence Relations
Everything you need for sequences: from calculating terms to long-term stability. Master limit theory and real-life modelling with these essential rules.
1
Foundational Formulas
The Recurrence Relation
\( u_{n+1} = au_n + b \)
Finding The Next Term
Multiply the current term by \( a \), then add \( b \).
Working Backwards
Subtract \( b \) from the next term, then divide by \( a \).
Percentage Multipliers
e.g. \( 4\% \) increase \( \to a = 1.04 \);
\( 4\% \) decrease \( \to a = 0.96 \).
\( 4\% \) decrease \( \to a = 0.96 \).
2
Limit Theory
Condition For A Limit
A sequence converges to a limit if and only if \( -1 < a < 1 \) (or \( |a| < 1 \)).
Important: Strict inequality is vital. \( a=1 \) or \( a=-1 \) does not converge.
Important: Strict inequality is vital. \( a=1 \) or \( a=-1 \) does not converge.
Limit Formula Calculation
\[ L = \frac{b}{1-a} \]
3
Problem Solving
Finding A and B
Use consecutive terms to create simultaneous equations.
Watch out: If terms are not consecutive (e.g. \( u_1, u_3 \)), this may form a quadratic equation.
Watch out: If terms are not consecutive (e.g. \( u_1, u_3 \)), this may form a quadratic equation.
Parameters & Convergence
Solve inequalities to find range of values of unknowns like k.
4
Contextual Meaning
Stable Levels
The limit \( L \) represents the "steady state" of chemicals, medicines, or populations over time.
Sequence Comparison
Determine if a value will "ever exceed" or "fall below" a threshold by comparing it directly to the limit \( L \).
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SQA Interpretation
- The Limit Mark: You MUST state that a limit exists because \( -1 < a < 1 \) to secure full marks.
- Communication: Always relate your final answer back to the context (e.g., "The population stabilizes at 14,286").
- Exact Values: Unless it is a calculator paper, keep fractions exact (e.g., \( \frac{22}{3} \)) rather than rounding.