# Recursively defined sequence worksheet

A recursive sequence is a sequence in which terms are defined using one or more previous terms which are given. Find the 9 th term of the arithmetic sequence if the common difference is 7 and the 8 th term is Example Names of standardized tests are owned by the trademark holders and are not affiliated with Varsity Tutors LLC.

Media outlet trademarks are owned by the respective media outlets and are not affiliated with Varsity Tutors. Varsity Tutors connects learners with experts. Instructors are independent contractors who tailor their services to each client, using their own style, methods and materials. Recursive Sequence A recursive sequence is a sequence in which terms are defined using one or more previous terms which are given.

Example 1: Find the 9 th term of the arithmetic sequence if the common difference is 7 and the 8 th term is Subjects Near Me. Download our free learning tools apps and test prep books. Varsity Tutors does not have affiliation with universities mentioned on its website.Consider recursive sequences with your class.

They solve 10 different problems that include various recursive formula sequences. First, they write a recursive formula for the sequence shown, then determine the first 4 terms of the sequence given the information. Save time and discover engaging curriculum for your classroom. Reviewed and rated by trusted, credentialed teachers. Get Free Access for 10 Days! Curated and Reviewed by. Lesson Planet. Reviewer Rating. More Less. Additional Tags. Resource Details. Grade 8th - 10th. Subjects Math 1 more Resource Types Worksheets 1 more Audiences For Teacher Use 1 more Start Your Free Trial Save time and discover engaging curriculum for your classroom.

Try It Free. Recursive Sequences Lesson Planet. Calculate the term, move to the next—and repeat. Pupils calculate the first five terms of a recursively defined sequence. They explain why the sequence is a function and describe its domain.

Arithmetic and Geometric Sequences Lesson Planet.

Recursive Definition of Sequences

Old mathematicians never die; they just lose some of their functions. Studying sequences gives scholars an opportunity to use a new notation.A sequence is recursively defined if its general term is determined using one or several of the terms preceding it. It is important to note that the first term or first couple terms must be given as part of the definition of the sequence. The Fibonacci Sequence is a perfect example of a recursively defined sequence as the general term is related to the previous terms.

See if you can identify the recursive definition for the Fibonacci Sequence. Value: 1 Which of the following gives the formula for the general term of the Fibonacci Sequence?

Recursively defined sequences are used often in computer programming and differential equations. It can sometimes be difficult to find the general term. Recall the following sequence:. What if we wanted to find a recursive relationship for the general term? Value: 1 Which of the following gives a recursive relationship for the nth term of the sequence. All Rights Reserved. Date last modified: August 12, Created by Dr. Phillip G. Lesson 12a Sequences and Series.

Recursively Defined Sequences A sequence is recursively defined if its general term is determined using one or several of the terms preceding it. Exampleis 3 and that each term is 2 times the one preceding it. Which of the following gives the formula for the general term of the Fibonacci Sequence?Teachers Pay Teachers is an online marketplace where teachers buy and sell original educational materials.

### Lesson 2 Recursive

Are you getting the free resources, updates, and special offers we send out every week in our teacher newsletter? All Categories. Grade Level. Resource Type. Log In Join Us. View Wish List View Cart. Results for recursive sequences Sort by: Relevance. You Selected: Keyword recursive sequences. Grades PreK. Other Not Grade Specific. Higher Education. Adult Education. Digital Resources for Students Google Apps. Internet Activities.

English Language Arts. Foreign Language. Social Studies - History. History World History. For All Subject Areas. See All Resource Types. Includes TWO options!Sections: Common differencesRecursions, General examplesNon-math "sequences". Take another look at the last sequence in the previous section :. The formula we found for the terms was a bit messy, what with the fractions.

But the row of first differences points out a simpler rule. Each next term was gotten by adding a growing amount to the previous term. To get the second term, they added 3 to the first term; to get the third term, they added 4 to the second term; to get the fourth term, they added 5 to the third term; and so on.

This sort of sequence, where you get the next term by doing something to the previous term, is called a "recursive" sequence. In the last case above, we were able to come up with a regular formula a "closed form expression" for the sequence; this is often not possible or at least not reasonable for recursive sequences, which is why you need to keep them in mind as a difference class of sequences.

Probably the most famous recursive sequence is the Fibonacci sequence pronounced "fibb - uh - NAH - chee" sequence. It is defined like this:. That is, the first two terms are each defined to have the value of 1. These are called "seed" values. And so forth. The first few terms are:.

## Lesson 2 Recursive

While recursive sequences are easy to understand, they are difficult to deal with, in that, in order to get, say, the thirty-nineth term in this sequence, you would first have to find terms one through thirty-eight. Well, there isbut its development is likely far beyond anything you've yet been trained to do. For instance, if you try to find the differences, you'll get this:. As you can see, you're not going to get a row of differences where all the entries are the same.

However, you should notice that the sequence repeats itself in the lower rows, but shifted over to the right. This is characteristic of "add the previous terms" recursive sequences. If you see this kind of behavior in the rows of differences, you should try finding a recursive formula. Recursive sequences can be hard to figure out, so generally they'll give you fairly simple ones of the "add a growing amount to get the next term" or "add the last two or three terms together" type:.

The first two values appear to be seed values, and then I'm adding the last two terms to get the next term:. If I'm right about the rule, then the next term would be:. By the way, the differences look like this:. Note how the sequence terms are repeated in lower rows, but shifted to the right, and how the new sequence terms are entering from the left.

This one is harder and is not, strictly speaking, recursive. Take a look at the differences:. As you can see, we're getting nothing useful from this table of differences. The rule here is not consistent:. Then the next term will be "add five":.Our printable recursive sequence worksheets provide ample practice for high school students on various topics like writing arithmetic sequence, geometric sequence and general sequence using the recursive formula, determining the recursive formula for the given sequences, finding the specific term and more.

Some of these worksheets are absolutely free of cost! Arithmetic Sequence. First term and the recursive formula are given in these pdf worksheets. Write the arithmetic sequence using the implicit formula.

Arithmetic Sequence - Recursive Formula. Using the first term 'a' and the common difference 'd', write the recursive formula for each arithmetic sequence. The terms of the sequence involve integers, decimals and fractions. Arithmetic Sequence - Mixed Review. Engage yourself with these printable arithmetic sequences mixed review worksheets and find the arithmetic sequence using the recursive formula in part A and vice versa in part B.

Geometric Sequence. Write the geometric sequence using the first term and the recursive formula. There are ten problems in each pdf worksheet for high school students.

Geometric Sequence - Recursive Formula. Observe the geometric sequence provided, identify the common ratio and find the recursive formula. Geometric Sequence - Mixed Review. This set of mixed review pdf worksheets contains exercises based on writing geometric sequence from recursive formula and vice versa. General Sequence. General sequence consists of numbers that are arranged in a particular order or pattern. Find the terms by plugging in the values of n in the recursive formula. General Sequence - Recursive Formula.

Determine the pattern followed in the sequence and write the recursive formula for nth term.

## How to Solve Recursive Sequences

Worksheets consist of 10 problems per page. Find the Indicated term. These printable worksheets require high school students to find the specific term of each sequence using the recursive formula. Problems are furnished in the word format. Members have exclusive facilities to download an individual worksheet, or an entire level.

Login Become a Member. Arithmetic Sequence First term and the recursive formula are given in these pdf worksheets.This extensive collection of series and sequence worksheets is recommended for high school students.

Explore various types of sequences and series topics like arithmetic series, arithmetic sequence, geometric sequence, finite and infinite geometric series, special series, general sequence and series, recursive sequence and partial sum of the series.

These arithmetic sequence worksheets comprise of an array of topics, like finding the arithmetic sequence, first term and common difference, general term of an arithmetic sequence, recursive formula and more. Gain access to this set of arithmetic series worksheets that requires students to evaluate arithmetic series, summation notation, determine the number of terms, real-life word problems and more.

Engage this collection of worksheets to practice finding geometric sequence, determining first term, common ratio, general term, next three terms and more! This assortment of finite geometric series worksheets includes topics like evaluating series, determine the number of terms, finding 'a' and 'n' and more! Get ample practice in the concept of infinite geometric series and learn to identify whether the series converges or diverges. Evaluate the sums of the infinite series. This collection of special series worksheets centralizes the concept of determining the sum of the series related to natural numbers and finding n th term for special series and much more! Learn to distinguish whether the sequence is arithmetic or geometric. This array of worksheets includes finding the explicit formula, finding the missing terms and much more. This batch of general series includes exercises like rewrite each series as an expanded sum, rewrite each series using sigma notation, evaluate the series and more.

These recursive sequence worksheets concentrate on the idea of finding the recursive formula for the given sequences and ascertain the sequence from the implicit formula provided. Partial sum of the series worksheets require students to determine the n th partial sum of the series, find the n th term of the series and are categorized based on the difficulty level. Login Become a Member. Arithmetic Sequence These arithmetic sequence worksheets comprise of an array of topics, like finding the arithmetic sequence, first term and common difference, general term of an arithmetic sequence, recursive formula and more.

Arithmetic Series Gain access to this set of arithmetic series worksheets that requires students to evaluate arithmetic series, summation notation, determine the number of terms, real-life word problems and more. Geometric Sequence Engage this collection of worksheets to practice finding geometric sequence, determining first term, common ratio, general term, next three terms and more! Finite Geometric Series This assortment of finite geometric series worksheets includes topics like evaluating series, determine the number of terms, finding 'a' and 'n' and more!

Infinite Geometric Series Get ample practice in the concept of infinite geometric series and learn to identify whether the series converges or diverges. Special Series This collection of special series worksheets centralizes the concept of determining the sum of the series related to natural numbers and finding n th term for special series and much more!

General Sequence Learn to distinguish whether the sequence is arithmetic or geometric. General Series This batch of general series includes exercises like rewrite each series as an expanded sum, rewrite each series using sigma notation, evaluate the series and more.

Recursive Sequence These recursive sequence worksheets concentrate on the idea of finding the recursive formula for the given sequences and ascertain the sequence from the implicit formula provided. Partial Sum of the Series Partial sum of the series worksheets require students to determine the n th partial sum of the series, find the n th term of the series and are categorized based on the difficulty level.

Arithmetic Sequence.