Arithmetic and geometric progressions in A Level Math Paper 1: common mistakes and examiner advice

Progressions questions are mostly about choosing the right formula, using the right value of n, and interpreting the context. The reports show that students who read the situation carefully do well, and that most lost marks come from mixing up the formulas or the position of a term. This guide collects what Cambridge examiners say students get wrong, based on recent reports.

Summarised in our own words from a reading of the examiner reports.

Mistakes that cost marks

• The wrong formula. Students used the formula for the sum when the nth term was needed, or the reverse. One report records the sum of 60 terms found instead of the 60th term. Some used arithmetic progression methods on a geometric progression.
• The wrong value of n. Common mistakes are using n instead of n − 1 in the nth term, using the wrong number of terms, or assuming n was unchanged from an earlier part. In one question about a yearly increase, the first term had to be 5.02 with n = 20 (or 5.00 with n = 21), and many students did not account for this.
• Forming the equation wrongly. Reports list: making the sum of the first 10 terms equal to the sum of the first 5 or 15 terms, forming an equation where a term is 36 less instead of 36 more, putting the 8 on the wrong side of the equation, multiplying the wrong side by 10, and confusing p and d.
• Accepting an impossible common ratio. Students found a value of r greater than 1 and used it in the formula for the sum to infinity, instead of recognising that an error had been made. The condition |r| < 1 is a check on your own work.
• Choosing between two solutions. When a quadratic gives two values, the one that satisfies the conditions has to be chosen. The most common error was to fail to identify the single valid solution.
• The wrong first term. Students assumed the first term was 9 when it was not, or gave the first term of a sequence as S₂ instead of S₂ − 1.
• Context. In real-life problems, students did not see that a quantity was decreasing, so the common difference had to be negative, or used 0.05 or 105 as a common ratio instead of 1.05. When an inequality is formed with a negative common difference, the inequality can be reversed by mistake.
• Summing term by term. Adding individual terms is slow and introduces calculation and rounding errors.
• Percentages. In a final part, the amount lost and the amount left were mixed up.

What examiners advise

• Think about the most efficient way to solve the equations. Reports say that complicated substitutions are more error-prone than elimination.
• Use a relationship between two expressions for r (or d) when a question gives you three terms.
• Look at the form of the question. A general term that is linear in n points to an arithmetic progression.
• If a result looks impossible, such as r > 1 for a sum to infinity, go back and check your working.
• Read the first part of the question's context again before assuming the same value of n applies.

Habits that help

• Write down a, d or r, n and what you are finding before choosing a formula.
• Write the first few terms to check that your formula gives them.
• Use words such as "the 10th term" and "the sum of the first 10 terms" in your working to avoid mixing them up.

Questions students ask

How do I know whether to use the nth term or the sum formula?

Write down a, d or r, n and exactly what you are finding before choosing a formula. Reports record the sum of 60 terms being found instead of the 60th term, and the reverse. Use phrases such as "the 10th term" and "the sum of the first 10 terms" in your working.

What value of n should I use?

Check the situation. Common mistakes are using n instead of n − 1 in the nth term, using the wrong number of terms, or assuming n is unchanged from an earlier part. In one yearly-increase question, the first term had to be 5.02 with n = 20 (or 5.00 with n = 21).

My common ratio is greater than 1 and I need the sum to infinity. What do I do?

Go back and check your working. The condition |r| < 1 is a check on your own work: students used an r greater than 1 in the formula instead of realising an error had been made.

A quadratic gave two values for r or d. How do I choose?

Use the conditions in the question to pick the one valid solution. Examiners say the most common error was failing to identify the single valid solution.

How do I use percentage changes in a geometric progression question?

The ratio is 1.05 for a 5% increase, not 0.05 or 105. In context questions, notice when a quantity is decreasing, because the common difference then has to be negative.