What you need to know
Sequences describe ordered terms; series add those terms. Reliable work starts by identifying whether the pattern is arithmetic, geometric or neither before choosing a formula.
- Identify common difference for arithmetic sequences.
- Identify common ratio for geometric sequences.
- Distinguish the nth term from the sum of the first n terms.
- For an infinite geometric series, check convergence before using the sum formula.
Key mathematics
Arithmetic nth term
u_n=a+(n-1)d
a is the first term and d is the common difference.Arithmetic sum
S_n=\frac n2\left(2a+(n-1)d\right)
Equivalent to n/2 times first plus last.Geometric nth term
u_n=ar^{n-1}
r is the common ratio.Worked example
For an arithmetic sequence with first term 7 and difference 4, $$u_n=7+4(n-1)=4n+3.$$ The 20th term is $83$. The sum of the first 20 terms is $$S_{20}=\frac{20}{2}(7+83)=900.$$
EXAM TECHNIQUE
Exam tips
- Write down a, d or r before substituting into formulas.
- Check whether the question asks for a term or a sum.
- For geometric sequences, use ratio between consecutive terms consistently.
WATCH OUT
Common mistakes
- Using n instead of n minus 1 in the nth-term formula.
- Using the arithmetic sum formula for a geometric sequence.
- Applying an infinite geometric sum when the absolute value of r is at least 1.
Frequently asked questions
How do I tell arithmetic from geometric?
Arithmetic sequences have a constant difference; geometric sequences have a constant ratio.
What is the difference between u n and S n?
u n is one term; S n is the sum of the first n terms.
When does an infinite geometric series converge?
When the absolute value of the common ratio is less than 1.