The sequence is arithmetic with first term a 1=3 and common difference d=10. To write a recursive rule for a sequence, the initial term(s) must be included. Use a table to organize the terms and find the pattern.Ī recursive equation for a sequence does not include the initial term. Recursive Equations for Arithmetic and Geometric SequencesĪ n= a ( n-1)+ d, where d is the common differenceĪ n= r⋅ a ( n-1), where r is the common ratio In general, rules for arithmetic and geometric sequences can be written recursively as follows. In part (b), the ratios of consecutive terms are constant, so the sequence is geometric. In part (a) of Example 3, the differences of consecutive terms of the sequence are constant, so the sequence is arithmetic.
![recursive formula for geometric sequence recursive formula for geometric sequence](http://i.ytimg.com/vi/IGFQXInm-co/maxresdefault.jpg)
Write the first six terms of each sequence.Ī. A recursive rule gives the beginning term(s) of a sequence and a recursive equation that tells how an is related to one or more preceding terms. In this section, you will learn another way to define a sequence-by a recursive rule. An explicit rule gives a n as a function of the term’s position number n in the sequence. So far in this chapter, you have worked with explicit rules for the nth term of a sequence, such as a n=3 n-2 and a n=7⋅(0.5) n. ► Use recursive rules to solve real-life problems. ► Translate between recursive and explicit rules for sequences. ► Evaluate recursive rules for sequences. The first three terms of the sequence are 2, 5, and 8. Write the first three terms of the sequence a n=3 n-1. Each term is a power of 2 with the exponent equal to the number of the term. Therefore, the next three terms should be 2 5, 2 6, and 2 7 or 32, 64, and 128.ī. It appears that each term of the sequence is a power of 2: 2 1, 2 2, 2 3, 2 4, …. Write a recursive definition for the sequence.Ī. List the next three terms of the sequence 2, 4, 8, 16, ….
![recursive formula for geometric sequence recursive formula for geometric sequence](https://mathbitsnotebook.com/Algebra2/Sequences/recSeqPic2.jpg)
Therefore, for each term after the first,Ĥ=½(8), 2=½(4), 1=½(2), 0.5=½(1), 0.25=½(0.5), 0.125=½(0.25), …įor n>1, we can write the recursive definition:Īlternatively, we can write the recursive definition as:Ī rule that designates any term of a sequence can often be determined from the first few terms of the sequence.Ī. In this sequence, each term after the first is ½ the previous term. The formula that allows any term of a sequence, except the first, to be computed from the previous term is called a recursive definition.įor example, the sequence that lists the heights to which a ball bounces when dropped from a height of 16 feet is 8, 4, 2, 1, 0.5, 0.25, 0.125, …. Most sequences are sets of numbers that are related by some pattern that can be expressed as a formula. The terms of a sequence are often designated as a 1, a 2, a 3, a 4, a 5, … If the sequence is designated as the function f, then f(1)= a 1, f(2)= a 2, or in general: In this case, the domain is the set of positive integers.
![recursive formula for geometric sequence recursive formula for geometric sequence](http://mrsnyderalgebra2.weebly.com/uploads/8/7/6/7/8767410/geometric-recursion-pic1_1_orig.png)
Often the sequence can continue without end. The function that lists the height of the ball after 7 bounces is shown on the graph at the right.
![recursive formula for geometric sequence recursive formula for geometric sequence](https://d2vlcm61l7u1fs.cloudfront.net/media/90d/90da6238-bbc2-41ca-ac15-cecfd644e865/phpncgw6Q.png)
Therefore, a sequence is a special type of function. We associate each term of the sequence with the positive integer that specifies its position in the ordered set. The general term is defined for n=1, 2, 3, 4, …. The general term or nth term of a sequence is represented by a symbol with a subscript, for example a n, t n, or u n. The dots must not be joined because n must be an integer.