Arithmetic series · Year 12
In Year 12 you learn: arithmetic sequences and series; sigma notation.
An arithmetic series is the sum of a sequence that goes up or down by the same amount, the common difference d, each time. Its sum is the number of terms times the average of the first and last terms.
Key facts
- Arithmetic: uₙ = a + (n − 1)d and Sₙ = n/2 (2a + (n − 1)d) = n/2 (a + l).
- Sigma notation: Σ uᵣ from r = 1 to n means u₁ + u₂ + … + uₙ.
- a is the first term, d the common difference and l the last term.
Worked examples
Example 1
Find the 20th term of the arithmetic sequence 5, 9, 13, …
Answer: 81
Example 2
Find the sum of the first 30 terms of 7 + 11 + 15 + …
Answer: 1,950
Example 3
Evaluate Σ (5r − 2) from r = 1 to 20.
Answer: 1,010
Example 4
An arithmetic series has first term 6 and 15th term 48. Find the sum of the first 15 terms.
Answer: 405