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Grade 11 / High 2 (age 16-17)

Recurrence Relations

Recurrence Relations

Build the Next Term from the Previous
5
👀 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
an+1 = an + d ⇒ arithmetic, an = a1 + (n−1)d
Adding a constant d gives an arithmetic sequence
Geometric-type recurrence
an+1 = r·an ⇒ geometric, an = a1·rn-1
Multiplying by a constant ratio r gives a geometric sequence
Difference Type — When the Added Amount Varies
Difference-type recurrence
an+1 = an + f(n) ⇒ an = a1 + ∑k=1n-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, an+1=an+2, find a5.
1
Adding a constant 2 means an arithmetic sequence with common difference 2.
an = 3 + (n−1)·2
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, an+1=2an+1, find a4.
1
Apply the rule term by term.
a2=3, a3=7, a4=15
2
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
an+1=an+d (arith), an+1=r an (geo), an+1=an+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, an+1=an+3, what is a10?
①26
②29
③32
④35
⑤38
▸ ② 29
1
It is an arithmetic sequence with common difference 3.
an = 2 + (n−1)·3
2
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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