SM1.5 Arithmetic progressions
You can recognise an arithmetic progression, find its nth term and the sum of its first n terms, and use both formulas inside word problems.
Topics shown in module order.
An arithmetic progression is a sequence that climbs or falls by the same step every time: instalments, rows of seats, salaries with a fixed yearly rise. Two formulas run the whole chapter, one for the nth term and one for the sum of the first n terms, and every question is one of the five quantities in them asked for from the others. The sum formula is the young Gauss trick, pairing the first term with the last, written down in general.
Work through these
Given the first few terms of several sequences, decide which are arithmetic progressions and find each common difference
Checking that the step is the same everywhere, not only between the first two terms, is the definition doing its work. Sequences that look regular but are not, like the squares, are the useful counterexamples.
Use the nth term formula in all directions: find a term, find which position a value sits at, find the first term or the step
The formula has four quantities and a question can hide any one of them. Practising every direction, including the one that answers with a position rather than a value, is what makes the formula an instrument rather than a chant.
Derive the sum formula once by writing the series forwards and backwards and adding
The derivation takes five lines and once seen is never forgotten: every pair adds to the same total. It also tells you when the formula applies, which memorising cannot.
Solve two word problems that combine the term and sum formulas
Real questions rarely ask one formula in isolation: how many instalments until a loan is cleared needs both. Combining them under a story is the chapter's finished form and the examination's favourite shape.
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