Sequences
Bus numbers, house numbers on a street, savings each month: lists in a definite order are everywhere. In maths such an ordered list is a sequence.
A sequence is a list of numbers in a definite order: a₁, a₂, a₃, …, with aₙ the nth term (general term). It is finite if it ends, infinite otherwise. A sequence is often given by a formula for aₙ, or by a recurrence that gives each term from the earlier ones.
Formula: a₁, a₂, a₃, …, aₙ
Rule 1: Terms from aₙ — Put n = 1, 2, 3, … in the formula.
- Easy: aₙ = 2n + 3: first terms 5, 7, 9, 11, 13.
- Medium: aₙ = n²: first terms 1, 4, 9, 16, 25.
- Exam-type: aₙ = (−1)ⁿ n: first terms -1, 2, -3, 4, -5.
Rule 2: Terms from a recurrence — Use the rule starting from the given first terms.
- Easy: a₁ = 3, aₙ = aₙ₋₁ + 4: 3, 7, 11, 15, 19.
- Medium: a₁ = 2, aₙ = 3aₙ₋₁: 2, 6, 18, 54, 162.
- Exam-type: Fibonacci: a₁ = a₂ = 1, aₙ = aₙ₋₁ + aₙ₋₂: 1, 1, 2, 3, 5, 8, 13, 21.
Order matters: 1, 2, 3 and 3, 2, 1 are different sequences.
Starting the count from n = 0 when the sequence starts at a₁.
Sequence = a list with a rule.
Practice questions
- 1Write the first 4 terms of aₙ = 3n − 1.
- 2Write the 10th term of aₙ = n²+ 1.
- 3Write the first 5 terms if a₁ = 5 and aₙ = aₙ₋₁ − 2.
- 4Find the 7th term of aₙ = n(n + 1)/2.
- 5Find which term of aₙ = 4n − 3 equals 37.
- 6Find the first 4 terms of aₙ = (−1)ⁿ⁺¹ 2ⁿ.
- 7Find a₆ if a₁ = 1 and aₙ = 2aₙ₋₁ + 1.
- 8For the Fibonacci sequence find the sum of the first 6 terms.
Show answers (try them yourself first)
- 12, 5, 8, 11
- 2101
- 35, 3, 1, −1, −3
- 428
- 54n = 40, n = 10
- 62, -4, 8, -16
- 763
- 820 (1+1+2+3+5+8)
Series and sigma notation
Adding the pocket money of 5 weeks gives a total. Adding the terms of a sequence gives a series.
If a₁, a₂, …, aₙ is a sequence, the expression a₁ + a₂ + … + aₙ is a series. Its sum is written with the sigma notation: Σₖ₌₁ⁿ aₖ. The sum of the first n natural numbers is n(n + 1)/2.
Formula: Σₖ₌₁ⁿ aₖ = a₁ + a₂ + … + aₙ
Rule 1: Expand and add — Replace k by 1, 2, …, n and add.
- Easy: Σₖ₌₁⁴ k = 1 + 2 + 3 + 4 = 10.
- Medium: Σₖ₌₁⁵ (2k + 1) = 3 + 5 + 7 + 9 + 11 = 35.
- Exam-type: Σₖ₌₁³ k² = 1 + 4 + 9 = 14.
Rule 2: Write with sigma — Find the pattern for the kth term.
- Easy: 2 + 4 + 6 + 8 = Σₖ₌₁⁴ 2k.
- Medium: 1 + 3 + 5 + 7 + 9 = Σₖ₌₁⁵ (2k − 1).
- Exam-type: Sum of the first 20 natural numbers = 20 × 21/2 = 210.
The sum of the first n odd numbers is n² (1 + 3 + 5 + 7 = 16).
Forgetting that the sum starts at k = 1, not k = 0.
Sigma means "add them all".
Practice questions
- 1Find Σₖ₌₁⁵ k.
- 2Find Σₖ₌₁⁴ (3k).
- 3Find Σₖ₌₁⁶ (k + 1).
- 4Write 5 + 10 + 15 + 20 using sigma notation.
- 5Find the sum of the first 50 natural numbers.
- 6Find the sum of the first 10 odd numbers.
- 7Find Σₖ₌₁⁴ k².
- 8Find 1 + 2 + … + 100 − (1 + 2 + … + 98).
Show answers (try them yourself first)
- 115
- 230
- 327
- 4Σₖ₌₁⁴ 5k
- 51275
- 6100
- 730
- 899 + 100 = 199