nth term and sum of progressions

What this topic tests, the mistakes that cost marks, and one exam-style question worked through step by step.

Key ideas and common mistakes

• To find earlier terms in an arithmetic progression when a later term and the common difference are given, subtract multiples of the common difference from the known term.
  Common mistake: Students often add instead of subtracting or use the wrong multiple.
• When relating sums of different numbers of terms in an arithmetic progression, set up and solve an equation using the series sum formula for each case.
  Common mistake: Students often forget to use the correct n for each sum or misapply the formula.
• To relate the first term and common difference using a condition on sums of terms in an arithmetic progression, substitute the sum formula for each case and equate, being careful to expand and collect like terms for a and d before solving.
  Common mistake: A common error is to substitute values or not fully expand the sums, leading to incorrect coefficients.
• Recognize that ln of powers (e.g., ln 8 = 3 ln 2) rewrites the sequence as an arithmetic progression in multiples of ln 2, so the common difference is found using these multiples, not the original numbers.
• When summing a range of terms in a sequence, the number of terms is last minus first plus one (inclusive counting).
  Common mistake: Students often forget the '+1' and get one term too few.
• When solving for the smallest integer n where a sum exceeds a value, if the quadratic solution is not whole, you must round UP, since rounding down gives a sum just below the target.
  Common mistake: Students often mistakenly round down or take the integer part.

Worked example

Try one yourself

13 exam-style questions on nth term and sum of progressions, each marked to the scheme.

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Questions are exam-style practice written to the IGCSE Additional Mathematics (0606) syllabus. Cambridge IGCSE is a registered trademark of Cambridge University Press & Assessment; MOOBON Axiom is independent and not affiliated with it.