D: Sequences and series: Understand and work with arithmetic sequences and series, including the formulae for nth term and the sum to n terms.

The idea

An arithmetic sequence increases or decreases by a fixed common difference, d. The nth term is a + (n-1)d, where a is the first term. The sum of the first n terms is n/2[2a + (n-1)d] or n/2(a + l), where l is the last term. To work with these, identify a and d from the sequence, substitute into the correct formula, and solve for the unknown. For example, in 5, 8, 11, 14, ... a=5 and d=3, so the 20th term is 5+19×3=62 and the sum of the first 20 terms is 20/2(2×5+19×3)=10(10+57)=670. Recognise when a problem asks for a term, a sum, or the number of terms, and use simultaneous equations if two terms are given.

What earns the marks

• Correctly identifies the first term a and common difference d from a given arithmetic sequence or from contextual information.
• Uses the nth term formula a + (n-1)d accurately to find a specified term or to form an equation when a term is known.
• Applies the sum formula S_n = n/2[2a + (n-1)d] or S_n = n/2(a + l) correctly, choosing the appropriate version for the information given.
• Solves problems that require finding n, a or d by setting up and solving linear equations, including checking that n is a positive integer.
• Interprets arithmetic sequences in context, such as savings, seating or simple interest, and communicates the final answer with correct units where appropriate.

Common mistakes and how to put them right

• Using n instead of n-1 in the nth term formula, which shifts every term by one place.

  Put it right by remembering the first term corresponds to n=1, so the multiplier of d is n-1.

• Confusing the nth term with the sum to n terms.

  Put it right by checking whether the question asks for a single term or a total, and use the appropriate formula.

• Forgetting to divide by 2 in the sum formula or misplacing the factor n.

  Put it right by writing the formula clearly and substituting carefully, or by using S_n = n/2(a + l) when the last term is known.

Exam tips

• Write down a and d explicitly before substituting into any formula; this reduces substitution errors and makes method clear.
• When a question gives two terms, form two equations using a + (n-1)d and solve simultaneously to find a and d.
• Check whether the answer should be an integer for n; if solving a quadratic gives a non-integer or negative value for n, reject it and re-read the problem.