Arithmetic sequences
An arithmetic sequence adds the same number each time. That number is the common difference, d.
| Sequence | First term a | d |
|---|---|---|
| 3, 7, 11, 15, … | 3 | 4 |
| 20, 14, 8, 2, … | 20 | −6 |
| 1.5, 2, 2.5, 3, … | 1.5 | 0.5 |
To check: subtract each term from the next. If the result is always the same, it’s arithmetic.
The general term
To find any term without listing them all:
tₙ = a + (n − 1)d
where a is the first term, d the common difference and n the term number. This is the general term.
Example: 3, 7, 11, … Find t₂₀.
t₂₀ = 3 + 19 × 4 = 79
Why n − 1? To get from the 1st term to the 20th, you add d nineteen times.
Finding the number of terms
Example: How many terms in 5, 8, 11, …, 92?
- 92 = 5 + (n − 1)(3)
- 87 = 3(n − 1)
- n − 1 = 29 → n = 30
Arithmetic series
An arithmetic series is the sum of the terms.
Sₙ = n(a + tₙ)/2 or Sₙ = n[2a + (n − 1)d]/2
The Gauss story: As a schoolboy, Gauss was told to add 1 to 100. He paired the numbers: 1 + 100, 2 + 99, … each pair = 101, and there are 50 pairs. Sum = 50 × 101 = 5,050. That’s exactly the formula: 100(1 + 100)/2.
Example: Sum of the first 30 terms of 5, 8, 11, …
t₃₀ = 92 → S₃₀ = 30(5 + 92)/2 = 1,455
Linear connection
An arithmetic sequence is a linear function on whole numbers: tₙ = dn + (a − d). Its graph is points on a straight line with slope d.
Real-life examples
- Seats in a theatre: 20 in row 1, 22 in row 2, … (d = 2)
- Saving $25 more each week
- Simple interest: a balance growing by the same amount each year
- Stacking cups: each adds the same height
Worked examples
Example 1: Find t₁₅ for 100, 93, 86, … → a = 100, d = −7 → 100 + 14(−7) = 2.
Example 2: A theatre has 18 seats in row 1 and 2 more in each row. How many seats in 25 rows? t₂₅ = 18 + 48 = 66 → S = 25(18 + 66)/2 = 1,050.
Example 3: Find the sum 2 + 4 + 6 + … + 200. n = 100 → S = 100(2 + 200)/2 = 10,100.
Common mistakes
- Using n instead of n − 1 in the general term.
- Getting the sign of d wrong for decreasing sequences.
- Confusing the sequence (list) with the series (sum).
Try it yourself
- Find d and t₁₀ for 6, 11, 16, …
- Find t₅₀ for −3, −1, 1, …
- How many terms in 7, 13, 19, …, 151?
- Sum the first 20 terms of 4, 9, 14, …
- Sum of 1 + 2 + … + 50
Answers: 1) d = 5, t₁₀ = 51 2) 95 3) 25 4) 1,030 5) 1,275
Quick recap
- An arithmetic sequence has a common difference d.
- General term: tₙ = a + (n − 1)d.
- Arithmetic series: Sₙ = n(a + tₙ)/2.