Recurrence relations

Japanese school year: Math B

What you learn

Learn methods for determining the general term of a sequence from recurrence relations connecting consecutive terms. This mathematical approach is vital for modeling discrete dynamic processes and analyzing algorithms. Familiarity with basic sequences, summation formulas, and equation-solving techniques will assist your progress.

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Key points

Formula to find the $n$ -th term $a_n$ ( $n \ge 2$ ) from first term $a_1$ by summing the differences $b_k = a_{k+1} - a_k$ between adjacent terms.

a_n = a_1 + \sum_{k=1}^{n-1} b_k \quad (n \ge 2)

Subtracting a constant $\alpha$ from both sides transforms $a_{n+1} = p a_n + q$ into a geometric sequence form to solve for $a_n$ .

a_{n+1} - \alpha = p(a_n - \alpha) \quad (\alpha = p\alpha + q)

Since the difference sequence formula applies only for $n \ge 2$ , always verify that the resulting formula also holds true for the first term $n = 1$ .

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Practice 1Coming soon