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Higher - Sequences
infoWhy this? Sequences develop our ability to recognise structure, generalise patterns, and calculate terms without listing every preceding value. Comparing linear, quadratic, geometric, and other sequences helps us distinguish different rules of change and express them precisely.
scheduleWhy now? Algebraic manipulation provides the tools for forming and using nth-term expressions, while recent index and surd work allows us to explore a wider range of terms. This also prepares us to understand iteration as the generation of successive approximations in the following equation-solving unit.
neurologyYou need to know
- A linear sequence increases or decreases by the same amount each time (constant difference).
- The nth term of a linear sequence has the form ( an + b ), where ( a ) is the common difference and ( b ) is the first term minus ( a ).
- Any term in a linear sequence can be found by substituting the value of ( n ) into the nth term expression.
- To determine if a number is a term in a linear sequence, you can solve the nth term formula for ( n ) and check if ( n ) is a positive integer.
- Fractional sequences are sequences where the terms are fractions, but the nth term can still be written in the form ( an + b ) where ( a ) and/or ( b ) are fractions.
- Non-linear sequences include quadratic sequences (where the second difference is constant), geometric sequences (where each term is multiplied by a constant), and special sequences such as triangular, square, cube, and Fibonacci numbers.
- The nth term of a quadratic sequence has the form ( an2 + bn + c ).
- The nth term of a geometric sequence has the form ( arn-1} ), where ( a ) is the first term and ( r ) is the common ratio.
- Surds are irrational roots (such as √2) that can appear in sequences
- The Fibonacci sequence is formed by adding the two previous terms to get the next term, starting from 0, 1 (or 1, 1).
rocket_launchYou must be able to
- Find the nth term of any linear sequence given a list of terms.
- Determine whether a given number is a term in a sequence by either continuing the sequence or solving the nth term formula for ( n ).
- Find the nth term of a fractional sequence.
- Generate the first five terms of a non-linear sequence when given the nth term, including for triangular, square, cube, Fibonacci, quadratic, and simple geometric sequences.
- Find any term in a non-linear sequence by substituting ( n ) into the nth term formula.
- Recognise and use sequences involving surds and other non-standard forms.
- Find the nth term of a quadratic sequence given a list of terms.
Revision Quiz
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