Difference Method for Finding Sequence Formulas: Principles and Step-by-Step Practice | 差分法求数列通项：原理与实战步骤

📚 Difference Method for Finding Sequence Formulas: Principles and Step-by-Step Practice | 差分法求数列通项：原理与实战步骤

The method of differences is a powerful way to find the n-th term of a sequence. When consecutive terms follow a polynomial pattern, subtracting them repeatedly eventually gives a constant row, and this constant row tells you the degree of the polynomial.

差分法是一种寻找数列第n项通项的强力方法。当相邻项呈现多项式规律时，反复作差最终会得到常数列，而这个常数列可以告诉我们多项式的次数。

1. What is the Method of Differences? | 1. 什么是差分法？

For a sequence a₁, a₂, a₃, …, the first difference is the result of subtracting a term from the next term. It is written as Δaₙ.

对于数列a₁, a₂, a₃, …，一阶差分是相邻两项相减的结果，记作Δaₙ。

Δaₙ = aₙ₊₁ − aₙ

First differences show how the sequence changes at each step. If they are all equal, the sequence is linear. If they are not equal, we can take their differences to get the second difference.

一阶差分展示数列每一步的变化情况。若它们全部相等，则数列是线性的；若它们不相等，可以继续求差，得到二阶差分。

The second difference is the difference between consecutive first differences.

二阶差分是相邻一阶差分之间的差。

Δ²aₙ = Δaₙ₊₁ − Δaₙ

2. First Differences: Linear Sequences | 2. 一阶差分：线性数列

If the first differences have a constant value d, then the sequence is arithmetic. Its n-th term can be written using the first term a₁ and the common difference d.

若一阶差分恒为d，则数列是等差数列，其通项可以用首项a₁和公差d表示。

aₙ = a₁ + (n − 1)d

For example, the sequence 7, 10, 13, 16 has first differences 3, 3, 3. Since d = 3 and a₁ = 7, the formula is aₙ = 3n + 4. You can check that n = 1 gives 7, n = 2 gives 10, and so on.

例如，数列7, 10, 13, 16的一阶差分都是3。因为d = 3且a₁ = 7，所以通项为aₙ = 3n + 4。可以验证n = 1时得到7，n = 2时得到10，以此类推。

3. Second Differences: Quadratic Sequences | 3. 二阶差分：二次数列

If the second differences are constant, the sequence is quadratic. Let the general form be aₙ = An² + Bn + C. For a quadratic expression, the second difference is always 2A.

若二阶差分恒定，则数列是二次数列。设通项形式为aₙ = An² + Bn + C。对于二次表达式，二阶差分恒等于2A。

if Δ²aₙ = c, then A = c ÷ 2

Consider the sequence 5, 9, 15, 23, 33. The first differences are 4, 6, 8, 10, and the second differences are 2, 2, 2. Since c = 2, we get A = 1.

观察数列5, 9, 15, 23, 33。一阶差分为4, 6, 8, 10，二阶差分为2, 2, 2。由于c = 2，可得A = 1。