Leave lesson

Pure · Sequences & series

1 / 8

Sigma notation and recurrence relations

Reading and evaluating sums written with Σ, and generating sequences from a recurrence relation — including spotting periodic behaviour.

Pure · Sequences & series

Sigma notation and recurrence relations

Reading and evaluating sums written with Σ, and generating sequences from a recurrence relation — including spotting periodic behaviour.

Why it works

Sigma notation is shorthand for "add these up". ∑r=1nf(r)\displaystyle\sum_{r=1}^{n} f(r) means substitute r=1,2,…,nr = 1, 2, \dots, n into f(r)f(r) and total the results. Two facts make these quick:
  • A constant summed is counted: ∑r=1nk=nk\displaystyle\sum_{r=1}^{n} k = nk (it appears
nn times, not once).
  • Sums split: ∑(2r+1)=2∑r+∑1\displaystyle\sum (2r + 1) = 2\sum r + \sum 1.
Often the terms form an arithmetic or geometric series, so you can use those sum formulae instead of adding by hand. Count the terms carefully: ∑r=1n\sum_{r=1}^{n} has nn terms; ∑r=3n\sum_{r=3}^{n} has n−2n - 2.

Recurrence relations define each term from the previous one(s), e.g. un+1=2un−1u_{n+1} = 2u_n - 1 with u1=3u_1 = 3. You generate the sequence step by step: u2=2(3)−1=5u_2 = 2(3) - 1 = 5, u3=2(5)−1=9u_3 = 2(5) - 1 = 9, and so on. Here un+1u_{n+1} just means "the term after unu_n".

Some recurrences settle into a repeating cycle — a periodic sequence. If un+1=4−unu_{n+1} = 4 - u_n with u1=1u_1 = 1, then u2=3u_2 = 3, u3=1u_3 = 1, u4=3,…u_4 = 3, \dots — it repeats every 22 terms (period 22), which lets you sum many terms by pairing them.