Convert an explicit formula to a recursive formula

Learning Objectives Define and differentiate between explicit and recursive formulas for sequences. Identify the initial term and the common difference or common ratio from an explicit formula. Write a recursive formula for an arithmetic sequence given its explicit formula. Write a recursive formula for a geometric sequence given its explicit formula. Explain the relationship between consecutive terms in a sequence using recursive thinking. Apply the conversion process to solve simple sequence problems. Have you ever noticed how some things grow or change in a predictable way, like the number of petals on a flower or the money in a savings account? 🌸💰 In this lesson, you'll learn about two ways to describe these patterns: explicit and recursive formulas. We'll f...

TermDefinitionExample SequenceAn ordered list of numbers that follow a specific pattern or rule.The sequence 2, 4, 6, 8, ... is a list of even numbers. TermEach individual number in a sequence.In the sequence 2, 4, 6, 8, ..., the number 2 is the 1st term, 4 is the 2nd term, and so on. Explicit FormulaA rule that allows you to find any term in a sequence directly, using its position (n) in the sequence. You don't need to know the previous term.For the sequence 2, 4, 6, 8, ..., an explicit formula is $T_n = 2 imes n$, where $T_n$ is the nth term. Recursive FormulaA rule that defines a term in a sequence by relating it to the previous term(s). You need to know the first term(s) to start.For the sequence 2, 4, 6, 8, ..., a recursive formula is $T_n = T_{n-1} + 2$, with $T_1 = 2$. This m...

Converting Arithmetic Explicit to Recursive Given an explicit formula for an arithmetic sequence: $T_n = T_1 + d imes (n-1)$, the recursive formula is: $T_n = T_{n-1} + d$, with $T_1 = \text{first term}$. To convert an explicit arithmetic formula, identify the first term ($T_1$) and the common difference ($d$). The recursive rule states that any term is the previous term plus the common difference. Remember to always state the first term for a recursive formula. Converting Geometric Explicit to Recursive Given an explicit formula for a geometric sequence: $T_n = T_1 imes r^{(n-1)}$, the recursive formula is: $T_n = T_{n-1} imes r$, with $T_1 = \text{first term}$. To convert an explicit geometric formula, identify the first term ($T_1$) and the common ratio ($r$). The recu...