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neurologyYou need to know

• A sequence is an ordered list of numbers, and each number in the list is called a term.
• The position of a term in a sequence is its term number, usually written as `n` (with `n` being a positive whole number).
• An arithmetic sequence has a constant difference between consecutive terms (you add or subtract the same number each time).
• In an arithmetic sequence with first term `u_1` and common difference `d`, the nth term can be written as `u_n = u_1 + (n-1)d`.
• A linear sequence is the same as an arithmetic sequence because its terms change by a constant amount.
• A decreasing linear (arithmetic) sequence has a negative common difference, so each term is smaller than the one before it.
• The nth term of a linear (arithmetic) sequence can be written in the form `an + b`, where `a` is the common difference.
• In the form `an + b`, the first term is found by substituting `n = 1`, giving `u_1 = a + b`.
• A geometric sequence has a constant ratio between consecutive terms (you multiply or divide by the same number each time).
• You can test for an arithmetic sequence by checking whether the differences between consecutive terms are equal.
• You can test for a geometric sequence by checking whether the ratio (next term divided by previous term) stays the same (when division makes sense).
• A non-linear sequence does not have a constant difference between consecutive terms, so it is not arithmetic.
• Non-linear sequences can still be continued by identifying a consistent rule (for example, using squares, cubes, alternating operations, or changing differences).
• A number is a term in a sequence if it appears when you continue the sequence correctly or when it is produced by the nth term rule for a whole-number value of `n`.
• If a sequence is given by an nth term rule, any term can be found by substituting the required term number `n` into the formula.
• To generate the first five terms from an nth term rule, substitute `n = 1, 2, 3, 4, 5` into the formula.
• For a linear decreasing sequence written as `an + b`, the value of `a` must be negative.
• If you solve `an + b = k` to test whether `k` is in the sequence, `k` is only a term if the solution for `n` is a positive whole number.

• Identify whether a sequence is arithmetic by calculating consecutive differences and checking they are constant.
• Identify whether a sequence is geometric by calculating consecutive ratios (where possible) and checking they are constant.
• Continue a given sequence accurately for several terms by applying its term-to-term rule consistently.
• Find missing terms in increasing or decreasing non-linear sequences by spotting and applying the pattern (for example, changing differences, alternating operations, or square/cube patterns) and checking it works for all shown terms.
• Find the nth term of a linear decreasing sequence by using the constant difference to get `a`, then substituting a known term (often `n = 1`) into `an + b` to find `b`.
• Generate the first five terms of a linear decreasing sequence from its nth term by substituting `n = 1` to `5` and simplifying correctly.
• Find any specified term of a linear decreasing sequence from its nth term by substituting the given value of `n` and evaluating accurately.
• Determine whether a number is a term in a sequence by either continuing the sequence to see if it appears, or (when an nth term is given) solving `an + b = k` and checking `n` is a positive whole number.