Progression questions in Additional Math combine the formulae for the nth term and the sum with equations, inequalities, and sometimes logarithms or trigonometry. The reports show that the usual cause of lost marks is setting up the problem: choosing the right formula and the right values. 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
- Misreading the information. Using the sum formula twice, assigning given values to the wrong parts of the formulae, using two formulae for terms instead of the sum formula, and not linking p and q to a and n. In one question, students multiplied the 6th term by 1.5 instead of the 3rd term.
- Mixing formulae. Confusing the sum to infinity with the required sum, attempting to use a + l, and stating S₁₀ and S₂₀ without equating them. Forgetting to multiply S₁₀ by 3 was also common.
- Setting up percentages. Problems setting up the initial equation, with 195% often seen, and 3(1 − 0.8ⁿ) becoming 3 − 2.4n.
- Finding a and r. Finding the common ratio and then forgetting to find the first term, and omitting the index at the next stage of the working.
- Finding the common difference. Omitting brackets, so that d = 8x + 2 − 5x − 4 was seen instead of d = 8x + 2 − (5x − 4).
- Sums of part of a progression. Subtracting the sum to 100 terms only (118 600 was the most common wrong answer), and writing a(1 − r³)/(1 − r) = 168 instead of a/(1 − r) − a = 168, forgetting to subtract the first term.
- Signs and brackets. The most common error in one part was a sign error when finding the sum of ten terms, with a negative common ratio and an essential bracket omitted.
- Convergence. Many identified the correct common ratio but did not give enough explanation, with too many incorrectly stating that sin θ < 1 (the condition is a two-sided one on the common ratio).
- Inequalities for n. Not expressing a quadratic inequality in solvable form or showing the method, not interpreting the solution (giving a non-integer value, or rounding down rather than up to the nearest integer), relying only on trial and improvement without enough justification, and giving a value such as n = 35 that came from rounding incorrect values.
- Equations. Solving r² + r − 1 = 0 incorrectly, or not discarding a root such as r = −3. Using the laws of indices or logarithms in an invalid way (for example solving 2^(k−1) = 48 from 2^(k+1) − 2^(k−1) = 48). Many did not deal with the logarithms correctly and ended up with a common ratio in terms of logarithms.
- "Show that." Writing down the given answer after omitting key steps, and not seeing that rⁿ is a common factor. Some divided through by a and made no further progress.
- Exact values. Fractions and terminating decimals are exact values and do not need to be rounded.
- Layout. Many candidates did not consider how they were setting out their solution, making their work difficult to follow.
What examiners advise
- Write down what you know (a, d or r, n, and the sum) in a list, then pick the formula that uses them.
- Show the method for an inequality, express it in solvable form, and interpret the result as the least integer that satisfies it.
- Use brackets when subtracting expressions, as in a common difference.
- Show all the key steps in a "show that" question, and use a common factor where there is one.
- Keep exact values, and reject roots with a reason.
- Set out your work clearly so that a reader can follow it.
Habits that help
- For a "sum from the (k+1)th term" question, write S_n − S_k explicitly.
- Check a ratio by dividing the second term by the first and the third by the second.
- After finding n, substitute n and n − 1 back into the condition to confirm the least value.
Questions students ask
How do I choose the right progression formula?
Write down what you know, a, d or r, n and the sum, in a list, then pick the formula that uses them. Students used the sum formula twice, assigned given values to the wrong parts, or confused the sum to infinity with the required sum.
How do I find the common difference?
Use brackets when subtracting expressions: d = 8x + 2 − (5x − 4), not 8x + 2 − 5x − 4.
How do I find the sum of part of a progression?
Take the difference of sums carefully. Subtracting only the sum to 100 terms was the most common wrong answer in one question.
How do I show that a geometric progression converges?
Give the condition |r| < 1 as a two-sided statement. Many identified the correct common ratio but did not give enough explanation, and incorrectly stated that sin θ < 1.
How do I solve an inequality for n?
Express it in solvable form, show the method, and interpret the result as the least integer that satisfies it. Students gave a non-integer value or rounded the wrong way.
How should I handle a "show that" question?
Show all the key steps and use a common factor such as rⁿ where there is one. Do not write down the given answer after omitting key steps.
Practice this topic
Ready to try some? Work through past paper questions on this topic one at a time. Attempt each question fully first, and open the mark scheme only afterwards.