Hey there! Great question—this is a super common point of confusion when first working through recurrence relations, so you’re definitely not alone in scratching your head over this. Let’s break down these two terms clearly, so you can spot the difference right away:

What’s the Order of a Recurrence Relation? #

The order refers to the difference between the largest and smallest indices in the recurrence. In simpler terms, it’s how many previous terms you need to know to calculate the current term.

• Example: For the recurrence a_n = 4a_{n-1} - 3a_{n-3}, the indices involved are n, n-1, and n-3. The largest minus smallest index is n - (n-3) = 3, so this is a 3rd-order recurrence.
• Another way to think about it: If you need the last 2 terms (a_{n-1} and a_{n-2}) to find a_n, that’s a 2nd-order relation.

What’s the Degree of a Recurrence Relation? #

The degree refers to the highest power of the sequence terms in the recurrence. It’s about how the terms are raised to exponents, not their indices.

• Example: For the recurrence (a_n)^3 + 2a_{n-1} = 0, the term a_n is raised to the 3rd power, so this is a 3rd-degree recurrence.
• If all sequence terms are to the first power (like a_n = 2a_{n-1} + n), then the degree is 1.

Are They the Same Thing? #

Short answer: No—they describe completely different aspects of a recurrence relation. The order is about how far back in the sequence you need to look, while the degree is about the algebraic power of the sequence terms themselves.

To tie this to the resources you referenced: Both materials make this distinction clear (even if phrased slightly differently). One focuses on index differences for order, and the other highlights term exponents for degree—neither uses the terms interchangeably.

内容的提问来源于stack exchange，提问作者Bhaskar