Convert a recursive formula to an explicit formula

Learning Objectives Define and differentiate between recursive and explicit formulas for sequences. Identify arithmetic and geometric sequences from their recursive definitions. Convert recursive formulas for arithmetic sequences into their explicit form. Convert recursive formulas for geometric sequences into their explicit form. Apply the explicit formula to calculate the limit of a sequence as n approaches infinity. Analyze the convergence or divergence of a sequence using its explicit form. How can we predict the millionth number in a pattern without calculating all 999,999 numbers before it? 🤔 Let's learn the mathematical shortcut! This tutorial teaches you how to transform step-by-step recursive formulas into powerful, direct explicit formulas. Mastering this co...

TermDefinitionExample SequenceAn ordered list of numbers, often following a specific pattern. Each number in the list is called a term.The sequence of even positive integers is 2, 4, 6, 8, ... Recursive FormulaA formula that defines each term of a sequence by relating it to the preceding term(s). It requires an initial term to start.a_n = a_{n-1} + 3, with a_1 = 5. This means 'to get any term, add 3 to the previous term'. Explicit FormulaA formula that calculates any term in a sequence directly from its position number (n), without needing to know the other terms.a_n = 3n + 2. To find the 100th term, you just plug in n=100: a_100 = 3(100) + 2 = 302. Limit of a SequenceThe value that the terms of a sequence approach as the term number 'n' gets infinitely large. It descr...

Explicit Formula for an Arithmetic Sequence a_n = a_1 + (n-1)d Use this when a sequence is defined recursively by adding a constant difference, 'd', in each step (e.g., a_n = a_{n-1} + d). 'a_1' is the first term. Explicit Formula for a Geometric Sequence a_n = a_1 * r^(n-1) Use this when a sequence is defined recursively by multiplying by a constant ratio, 'r', in each step (e.g., a_n = r * a_{n-1}). 'a_1' is the first term. Limit of a Geometric Sequence lim_{n->∞} a_1 * r^(n-1) = { 0 if |r| < 1; a_1 if r = 1; DNE if r = -1 or |r| > 1 } This rule is a powerful shortcut to determine the limit of a geometric sequence once you've found its explicit formula. The outcome depends entirely on the common ratio 'r&#0...