Find recursive and explicit formulas

Learning Objectives Define and differentiate between a recursive formula and an explicit formula for a sequence. Derive the explicit formula for an arithmetic or geometric sequence given its first few terms or its recursive formula. Write the recursive formula for an arithmetic or geometric sequence. Connect the concept of a sequence formula to the concept of a limit at infinity. Use an explicit formula to calculate the limit of a sequence as n approaches infinity. Determine if a sequence converges or diverges based on its limit. Imagine a bouncing ball that gets 80% of its previous height with each bounce. How can we predict its height on the 10th bounce, and what happens to the height eventually? 🤔 This tutorial bridges the gap between sequences and calculus. You will le...

TermDefinitionExample SequenceAn ordered list of numbers, called terms, that follow a specific pattern or rule. We often denote the n-th term as a_n.The sequence of even positive integers is 2, 4, 6, 8, ... where a_1 = 2, a_2 = 4, and so on. Explicit FormulaA rule that defines the n-th term of a sequence, a_n, as a function of its position, n. It allows you to calculate any term directly without knowing the previous terms.For the sequence 2, 4, 6, 8, ..., the explicit formula is a_n = 2n. To find the 50th term, you just calculate a_50 = 2 * 50 = 100. Recursive FormulaA rule that defines the n-th term of a sequence, a_n, based on one or more preceding terms (e.g., a_{n-1}). It must always include a starting value (e.g., a_1).For the sequence 2, 4, 6, 8, ..., the recursive formula is a_n =...

Arithmetic Sequence Formulas Recursive: a_n = a_{n-1} + d, with a_1 given. Explicit: a_n = a_1 + (n-1)d. Use for sequences with a common difference, d, between consecutive terms. The explicit formula is essential for finding the limit of the sequence. Geometric Sequence Formulas Recursive: a_n = a_{n-1} * r, with a_1 given. Explicit: a_n = a_1 * r^(n-1). Use for sequences with a common ratio, r, between consecutive terms. The limit depends on the value of r. Limit of a Geometric Sequence For a sequence a_n = a_1 * r^(n-1), the limit lim_{n->∞} a_n is 0 if |r| < 1, and it diverges if |r| > 1. If r=1, the limit is a_1. This is a critical shortcut for determining the convergence of geometric sequences. It connects the explicit formula directly to the long-term...