# Explicit and Recursive Sequences or Formulas Worksheets

When we come across a sequence if we can understand the nature of the formula it is based upon it makes it much easier to work with an manipulate to be able to find terms. A sequence is a set of data that commonly follows an ascending pattern, but if they can descend as well. The formula that these sequences are based upon often come in two flavors. The formulas can be recursive where we can find the value of a fixed term within the sequence based on the term that is just before it. If the formula allows use to determine the value of term based on the position it is located within the set, we call this explicit. These worksheets will help students identify and understand the use of both explicit expressions and recursive formulas.

### Aligned Standard: HSF-BF.A.1a

- Basic Sequences Step-by-step Lesson- You are given a 2s sequence and asked to provide two formats of the process.
- Guided Lesson - If you understand the terms that are being presented, this is not that difficult.
- Guided Lesson Explanation - Make sure to check inverses when you are working with recursive formulas.
- Practice Worksheet - We switch between the formats. A good idea is to quiz them on the format differences as well.
- Matching Worksheet - See if you can match the parts of each format with the other.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

Start with the recursive or explicit formula and find the other.

- Homework 1 - Write an explicit and recursive formula for the following sequences.
- Homework 2 - A recursive formula is a something that we can use to determine the next term in a set or number sequence. It tells us how each term is connected to the next term.
- Homework 3 - Given the recursive formula, write the explicit formula for the sequence.

### Practice Worksheets

Explicit formulas give you the direct answer. The recursive formula gives you the next value.

- Practice 1 - Write an explicit and recursive formula for the following sequences: 1, 3, 5, 7...
- Practice 2 - Create both forms for: t
_{1}= 0 and t_{n}= t_{n-1}- 8 - Practice 3 - These are in the form of an equation.

### Math Skill Quizzes

Work off what the problem gives you.

- Quiz 1 - These are integer sequences.
- Quiz 2 - More of a systematic approach is required here.
- Quiz 3 - Represent these explicit formulas in other ways.

### What Are Explicit and Recursive Sequences or Formulas?

These types of formulas are continued patterns that involve the process of addition. They can also be presented as subtraction when adding negative numbers. When you see these patterns, they will often be presented in a confusing manner. They each have their own purposes and if we understand the nature of the sequence that is being presented to us, we can easily determining terms of interest found within the sequence.

**Explicit Expressions** - Have you seen a stop sign? It tells us to put a halt or stop driving at fixed point. It clearly explains what we need to do, and we can instantly make an adjustment. Similar to that, explicit expressions clearly explain or are expressed clearly. Hence, they are quite easy to understand and apply. To be more precise, explicit expressions are functions that are written in terms of input or independent variables. These are formulas where we can find the value of a specific term within the sequence based on its position.

**Recursive Processes** - Let's say you are given a sequence that represents numbers 1, 2, 4, 8, 16, and so on. We can easily figure out that there is a pattern being followed here, which is the original number being doubled that produces the next number. In these types of formulas, we can determine the value of a specific term based on the term just before it. The method of applying the formula over and over is known as a recursive process.