STEP-BY-STEP LESSON · §5.6
Defining Sequences Recursively
Compute a recursive sequence from its initial conditions.
Before you begin
Sequences
A sequence assigns one value to each permitted index. Whether it starts at 0 or 1 is part of its definition.
a_n = 2n + 1, n ≥ 0
In a sum, the index takes every integer value from the lower bound through the upper bound.
∑_{i=0}^{2} a_i = a_0 + a_1 + a_2Symbols
- a_(n−1)
- previous term
- n
- an integer index; each rule states its starting index
Definitions and notation for this topic
Recurrence relation
A rule expressing terms in terms of preceding terms.
The initial value and the range where the rule applies belong to the definition. This recurrence is an example, not a universal symbol for recurrence relations.
Separate glossary example
a₁=3 and a_n=a_(n−1)+2 for integers n≥2 define 3,5,7,9,… .
a₁=2; aₙ=aₙ₋₁+2 for n≥2.
Recursive and explicit sequence formulas
A recursive rule defines later terms using earlier ones and initial values. An explicit formula gives a_n directly in terms of n and fixed parameters.
Check an explicit candidate against both the recurrence and the initial values. Matching a few terms alone does not prove the formulas agree for every index.
Separate glossary example
a₀=3, a_n=a_(n−1)+2 for n≥1 describes the same sequence as a_n=3+2n for n≥0.
Step by step
Step 1 / 6
Find the initial data
A recurrence relates terms but does not by itself specify where to start.
a_0=2
Different initial values produce different sequences.
Worked example
a_0=2, a_n=2a_(n−1)+1. Find a_3.
- a_1=2·2+1=5, using the initial value.
- a_2=2·5+1=11; now the previous term is 5.
- a_3=2·11+1=23.
Optional self-check
If b_0=4 and b_n=b_(n−1)+3, what is b_2?
Show answer
10: first b_1=7, then b_2=10.