Recurrence Relations

A recurrence relation fixes an entire sequence from one starting term and a rule that builds each term from the one before. Give it a first term and a rule, and repeating that rule spins out the next terms without end. Add a fixed amount for an arithmetic sequence, multiply by a fixed ratio for a geometric one, and if the added amount changes, a sigma sum builds the general term. Here a slider adds terms one at a time so you can watch the recurrence draw the sequence.

Build the Next Term from the Previous

Terms generated n5

👀 See It

①A recurrence defines a sequence by a first term and a rule "next = function of previous"

②Starting from a₁ and applying the rule repeatedly fixes every term

③This is the recursive (inductive) definition of a sequence

Arithmetic and Geometric Types

Arithmetic-type recurrence

a_n+1 = a_n + d ⇒ arithmetic, a_n = a₁ + (n−1)d

Adding a constant d gives an arithmetic sequence

Geometric-type recurrence

a_n+1 = r·a_n ⇒ geometric, a_n = a₁·r^n-1

Multiplying by a constant ratio r gives a geometric sequence

Difference Type — When the Added Amount Varies

Difference-type recurrence

a_n+1 = a_n + f(n) ⇒ a_n = a₁ + ∑_k=1^n-1 f(k)

If the added amount varies with n, find the general term by that sum (sigma)

Compute It Directly

Example 1

For a₁=3, a_n+1=a_n+2, find a₅.

Adding a constant 2 means an arithmetic sequence with common difference 2.

a_n = 3 + (n−1)·2

Substitute n=5.

a₅ = 3 + 4·2 = 11

▸ 11

For a "previous + constant" form, just write the arithmetic general term directly.

Example 2

For a₁=1, a_n+1=2a_n+1, find a₄.

Apply the rule term by term.

a₂=3, a₃=7, a₄=15

Compute the terms directly to get a₄.

a₄ = 2·7 + 1 = 15

▸ 15

If the rule is neither arithmetic nor geometric, generating small terms directly is fastest.

Wrap-up

Key result

a_n+1=a_n+d (arith), a_n+1=r a_n (geo), a_n+1=a_n+f(n) (diff)

Read the recurrence form first to tell arithmetic from geometric from difference type

2020 CSAT Math type, adapted

For the sequence defined by a₁=2, a_n+1=a_n+3, what is a₁₀?

①26

②29

③32

④35

⑤38

▸ ② 29

It is an arithmetic sequence with common difference 3.

a_n = 2 + (n−1)·3

Substitute n=10.

a₁₀ = 2 + 9·3 = 29

🎯 Exam Points

①Identify the form first: +d is arithmetic, ×r is geometric

②Difference type a_{n+1}=a_n+f(n) uses sigma

③If stuck, generate a few terms from a_1 to guess the rule

④The first term and the rule fully determine the sequence

⑤After finding the general term, substitute the index carefully

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Mathematical Induction

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