Convert between explicit and recursive formulas

Learning Objectives Identify patterns in simple arithmetic sequences. Define and differentiate between explicit and recursive formulas. Write an explicit formula for a given arithmetic sequence. Write a recursive formula for a given arithmetic sequence. Convert an explicit formula into its equivalent recursive formula. Convert a recursive formula into its equivalent explicit formula. Have you ever noticed how some things grow or change in a predictable way? 📈 Like the number of steps you take each day if you add 100 more than the day before, or the money in your piggy bank if you add the same amount every week! In this lesson, you'll learn two cool ways to describe these patterns using 'formulas': explicit and recursive. Understanding these helps us predict...

TermDefinitionExample SequenceAn ordered list of numbers, where each number is called a 'term'.The sequence 3, 6, 9, 12, ... is a list of numbers in a specific order. TermEach individual number in a sequence.In the sequence 3, 6, 9, 12, ..., the number 3 is the 1st term, 6 is the 2nd term, and so on. PatternThe rule or relationship that connects the terms in a sequence, showing how to get from one term to the next.In the sequence 3, 6, 9, 12, ..., the pattern is 'add 3' to get the next term. Common Difference (d)The constant amount that is added or subtracted to get from one term to the next in an arithmetic sequence.In the sequence 3, 6, 9, 12, ..., the common difference (d) is 3 because you add 3 each time. Explicit FormulaA rule that allows you to find *any* term in...

Arithmetic Explicit Formula $a_n = a_1 + (n-1)d$ Use this formula to find any term ($a_n$) in an arithmetic sequence directly. You need to know the first term ($a_1$), the term's position ($n$), and the common difference ($d$). This formula is great for jumping far ahead in a sequence. Arithmetic Recursive Formula $a_n = a_{n-1} + d$ (and $a_1$ must be stated) Use this formula to find the next term ($a_n$) in an arithmetic sequence by adding the common difference ($d$) to the previous term ($a_{n-1}$). You *must* also state the first term ($a_1$) for the sequence to begin. Converting Explicit to Recursive 1. Find $a_1$ by plugging $n=1$ into the explicit formula. 2. Identify the common difference ($d$) from the explicit formula (it's the number multiplied by...