Sequences and Recurrence Relations

📋 Course Outline

1. Sequence Notation and Recurrences
2. Arithmetic Sequences
3. Arithmetic Sums
4. Geometric Sequences
5. Applications of Geometric Growth

📖 1. Sequence Notation and Recurrences

★ Must-know

📌 A sequence can be defined explicitly by a formula such as $u_n=\frac{n}{n+1}$ or recursively by giving initial term(s) and a relation connecting consecutive terms.

• To calculate successive terms of a recursively defined sequence, start from the given initial term(s), substitute them into the recurrence, and repeat the relation until the requested index is reached.

• A recurrence model translates a verbal evolution into algebra by identifying the previous value, the next value, and any multiplication, addition, percentage change, or averaging operation between them.

Further detail

• The recurrence $u_{n+1}=2u_n-1$ with $u_0=5$ defines a sequence whose first terms are obtained successively from the preceding term.

💡 Memory Hook

Explicit formula gives a term directly, whereas recurrence builds it from previous terms.

📖 2. Arithmetic Sequences

🔑 Key Concepts & Definitions

• Arithmetic sequence : a sequence for which the same number r, called the common difference, is added to each term to obtain the next term.