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Foundation - Sequences
infoWhy this? Sequences develop our ability to recognise patterns, distinguish different types of change, and express general relationships. Finding linear nth-term rules and using given non-linear rules lets us calculate distant terms efficiently and test whether values belong to a sequence.
scheduleWhy now? With algebraic substitution and numerical powers already established, sequences connect numerical patterns with expressions. This prepares us to relate inputs and outputs in function machines and graphs, and to distinguish additive from multiplicative change in later interest calculations.
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.
- 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, including those with fractional terms.
- Generate the first five terms of any linear sequence when given the nth term.
- Find any term in any linear sequence by substituting a value for ( n ) into the nth term formula.
- Determine whether a given number is a term in a sequence by either continuing the sequence or by solving the nth term formula for ( n ).
- Generate the first five terms of a non-linear sequence when given the nth term, including for triangular, square, cube, quadratic, geometric, and Fibonacci sequences.
- Find any term in a non-linear sequence by substituting a value for ( n ) into the nth term formula.
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