This module covers arithmetic progressions, where each term adds a fixed difference d, and geometric progressions, where each term multiplies by a fixed ratio r. You learn to find terms, add them up, and decide when an infinite sum exists. The common thread is that two pieces of information fix the whole sequence, and most questions are about turning words into those two equations.
Check the current Cambridge Additional Mathematics 0606 syllabus for the exact content in your exam year, including which formulae are given and the calculator rules. Our Additional Mathematics learning guide shows where this module sits among the others.
What should you already know?
You need to solve simultaneous linear equations, expand and solve simple quadratics, and work with indices such as 26 and (2/3)2. If quadratic work feels shaky, revise quadratic structure and discriminants first, because the last lesson ends in a quadratic inequality. Indices and logarithms in exponential and logarithmic reasoning help when n is the unknown in a geometric problem.
The key formulae
For an arithmetic progression, the nth term is a + (n − 1)d and the sum of n terms is n/2 (2a + (n − 1)d). For a geometric progression, the nth term is arn−1 and the sum of n terms is a(1 − rn)/(1 − r). If |r| < 1, the sum to infinity is a/(1 − r).
An orienting example
An arithmetic progression has 2nd term 9 and 6th term 25. Find a, d and the sum of the first 10 terms.
Step 1, equations: a + d = 9 and a + 5d = 25.
Step 2, subtract: 4d = 16, so d = 4. Then a = 9 − 4 = 5.
Step 3, sum: S10 = 10/2 × (2(5) + 9(4)) = 5 × 46 = 230.
Check: the terms are 5, 9, 13, 17, 21, 25, 29, 33, 37, 41 and they add to 230.
The same pattern of “conditions to equations, solve, then answer” returns in every lesson.
In which order should you study it?
- Find a term from two sequence conditions: the core move, forming two equations in a and d or r.
- Calculate a finite arithmetic sum: the sum formula, and counting the number of terms correctly.
- Recover a common ratio from terms: geometric progressions, roots, and the case of two possible ratios.
- Use an infinite sum only when its condition holds: the test |r| < 1 before any formula.
- Solve a combined term-and-sum problem: mixes the skills and ends with an inequality in n.
Then work through the mixed practice set. One lesson a day and the practice set at the weekend is a steady pace. The sequence and series laboratory lets you check your own examples along the way.
Which traps catch most students here?
- Using n instead of n − 1, so the 5th term becomes a + 5d.
- Miscounting the number of terms, dividing the span by d and forgetting to add one.
- Dropping a negative ratio when r2 or another even power is involved.
- Using a/(1 − r) without checking |r| < 1, and accepting a negative sum for positive terms.
- Treating a sum equal to the target as “greater than” the target, so the least n is one too small.
Each lesson shows one of these slips in full and then corrects it.
How should you use the practice set?
Attempt each question on paper, without opening the answers. Check every result against the original conditions, because a substitution check takes under a minute and catches most errors.
Record the question numbers you missed and use the routing list at the end of the practice set. If the same trap appears more than once, return to that lesson before trying new questions. The mistake log and retest queue is a simple way to track this.
Where does tuition fit?
Sequences can be learned well from lessons like these, and some students only need the practice. Others follow every step but stall when a question is worded in an unfamiliar way. Our teachers look at how you translate wording into equations, which is usually where the difficulty lives, in online one-to-one Additional Mathematics tuition.
After this module, the next topic is binomial expansion, which also uses patterns in powers and coefficients.