Study Notes
Sequences and series involve understanding patterns and sums of numbers in specific orders. Key concepts include arithmetic and geometric progressions, as well as the use of sigma notation and recurrence relations.
- Arithmetic sequence — A sequence where each term is obtained by adding a fixed number to the previous term. Example: 2, 4, 6, 8, where the common difference is 2.
- Arithmetic series — The sum of the terms in an arithmetic sequence. Example: The sum of the first 4 terms of 2, 4, 6, 8 is 20.
- Geometric sequence — A sequence where each term is obtained by multiplying the previous term by a fixed number. Example: 3, 6, 12, 24, where the common ratio is 2.
- Geometric series — The sum of the terms in a geometric sequence. Example: The sum of the first 3 terms of 3, 6, 12 is 21.
- Sum to infinity — The sum of an infinite geometric series that converges. Example: For a series with a common ratio of 0.5, the sum to infinity is 2.
- Sigma notation — A way to write the sum of a sequence using the Greek letter Σ. Example: Σ from i=1 to n of i represents the sum of the first n natural numbers.
- Recurrence relations — A way to define a sequence where each term is a function of the previous term. Example: un+1 = 2un + 3, starting with u1 = 6.
- Modelling with series — Using sequences and series to represent real-life situations. Example: Calculating total profits over time with a geometric series.
Exam Tips
Key Definitions to Remember
- Arithmetic sequence: A sequence with a constant difference between terms.
- Geometric sequence: A sequence with a constant ratio between terms.
- Sum to infinity: The sum of an infinite geometric series that converges.
Common Confusions
- Mixing up arithmetic and geometric sequences.
- Forgetting that the sum to infinity only applies when the common ratio is between -1 and 1.
Typical Exam Questions
- What is the nth term of an arithmetic sequence with a first term of 5 and a common difference of 3? Answer: 5 + (n-1) * 3
- How do you find the sum of the first 10 terms of a geometric sequence with a first term of 2 and a common ratio of 3? Answer: Use the formula Sₙ = a(1-rⁿ)/(1-r) where a = 2, r = 3, n = 10.
- What is the sum to infinity of a geometric series with a first term of 4 and a common ratio of 0.5? Answer: 8
What Examiners Usually Test
- Ability to derive and use formulas for nth terms and sums of sequences.
- Understanding of when a geometric series converges and how to calculate its sum to infinity.
- Application of sequences and series in real-life modelling problems.