Recursive Sequence

A recursive sequence is a sequence in which terms are defined using one or more previous terms which are given.

If you know the n^th term of an arithmetic sequence and you know the common difference , d , you can find the (n+1)^th term using the recursive formula a_n+1=a_n+d .

Example 1:

Find the 9^th term of the arithmetic sequence if the common difference is 7 and the 8^th term is 51 .

a₉=a₈+d

a₉=51+7=58

If you know the n^th term and the common ratio , r , of a geometric sequence , you can find the (n+1)^th term using the recursive formula. a_n+1=a_n⋅r .

Example 2:

Write the first four terms of the geometric sequence whose first term is a₁=3 and whose common ratio is r=2 .

How recursive formulas work

Recursive formulas give us two pieces of information:

1. The first term of the sequence
2. The pattern rule to get any term from the term that comes before it

Here is a recursive formula of the sequence 3,5,7,… along with the interpretation for each part.

Cool! This formula gives us the same sequence as described by 3,5,7,…

Writing recursive formulas

Suppose we wanted to write the recursive formula of the arithmetic sequence 5,8,11,…

The two parts of the formula should give the following information:

• The first term (left parenthesiswhich is 5 )
• The rule to get any term from its previous term (left parenthesiswhich is “add 3 “)

Therefore, the recursive formula should look as follows:

Arithmetic Sequence Recursive Formula

Before going to learn the arithmetic sequence recursive formula, let us recall what is an arithmetic sequence. It is a sequence of numbers in which every successive term is obtained by adding a fixed number to its previous term. For example, -1, 1, 3, 5, … is an arithmetic sequence as every term is obtained by adding a fixed number 2 to its previous term. This fixed number is usually known as the common difference and is denoted by d. Let us learn the arithmetic sequence recursive formula along with a few solved examples.

What Is Arithmetic Sequence Recursive Formula?

Recursion in the case of an arithmetic sequence is finding one of its terms by applying some fixed logic on its previous term. As we learned in the previous section that every term of an arithmetic sequence is obtained by adding a fixed number (known as the common difference, d) to its previous term. Thus, the arithmetic sequence recursive formula is:

Arithmetic Sequence Recursive Formula

The arithmetic sequence recursive formula is:

a_n=a_n−1+d

where,

• a_n = n^th term of the arithmetic sequence.
• a_n−1 = (n – 1)^th term of the arithmetic sequence (which is the previous term of the n^th term).
• d = The common difference (the difference between every term and its previous term).

Examples Using Arithmetic Sequence Recursive Formula

Example 1: Find the recursive formula of the arithmetic sequence 1, 5/4, 3/2, 7/4, …. using the arithmetic sequence recursive formula.

Solution:

The common difference is,

d = 5/4 – 1 (or) 3/2 – 5/4 (or) 7/4 – 3/2 = 1/4.

Thus, the recursive formula of the given arithmetic sequence is,

a_n=a_n−1+d

a_n=a_n−1+1/4

Answer: an=a_n−1+1/4

Example 2: Find the first 5 terms of an arithmetic sequence whose recursive formula is a_n=a_n−1−3 and a₁=−1.

Solution:

Example 3:

FAQs on Arithmetic Sequence Recursive Formula

What Is Arithmetic Sequence Recursive Formula?

The arithmetic sequence recursive formula is used to find a term of an arithmetic sequence by adding its previous term and the common difference. By this formula the n^th term of an arithmetic sequence a1, a2, a3, … whose common difference is ‘d’ is, an=an−1+da.

How To Derive Arithmetic Sequence Recursive Formula?

We know that the difference (common difference, d) between every two successive terms of an arithmetic sequence is always constant. i.e., any term (n^th term) of an arithmetic sequence is obtained by adding the common difference (d) to its previous term ((n – 1)^th term). i.e., the recursive formula of the given arithmetic sequence is, a_n=a_n−1+d.

What Is the Use of Arithmetic Sequence Recursive Formula?

The arithmetic sequence recursive formula is used to find any term of an arithmetic sequence without actually knowing the first term of the sequence. By this formula, an=an−1+d , where a_n is the n^th term , a_n−1 is the (n – 1)^th term, and d is the common difference (the difference between every term and its previous term).

What Is the n^th Term of the Sequence -4, 2, 8, 16, … Using Arithmetic Sequence Recursive Formula?

The common difference of the given sequence is, d = 2 – (-4) (or) 8 – 2 (or) 16 – 8 = … = 6.

By arithmetic sequence recursive formula, the n^th term is,

a_n=a_n−1+d

a_n=a_n−1+8

What Is A Sequence

Formally, a sequence is an enumerated collection of objects, but informally, a sequence is a countable structure representing an ordered list of elements or numbers.

Recursive Formula Definition

So, what is recursion?

A recursive definition, sometimes called an inductive definition, consists of two parts:

1. Recurrence Relation
2. Initial Condition

A recurrence relation is an equation that uses a rule to generate the next term in the sequence from the previous term or terms. In other words, a recurrence relation is an equation that is defined in terms of itself.

And all recurrence relations must come with an initial condition, which is a list of one or more terms of the sequence that precede the first term where the recurrence relation starts.

Example

For instance,

Notice that this looks just like the procedure we use for mathematical induction!

The idea behind inductive proofs is similar to a staircase, as the only way to reach the top is to climb all the steps before it, as noted by Math Bits. The same thing is happening with recursion – each step is generated from the step or steps preceding.

Recursive Formulas For Sequences

Alright, so as we’ve just noted, a recursive sequence is a sequence in which terms are defined using one or more previous terms along with an initial condition. And the most classic recursive formula is the Fibonacci sequence.

The Fibonacci sequence is as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21,…

Notice that each number in the sequence is the sum of the two numbers that precede it. For example, 13 is the sum of 5 and 8 which are the two preceding terms.

In fact, the flowering of a sunflower, the shape of galaxies and hurricanes, the arrangements of leaves on plant stems, and even molecular DNA all follow the Fibonacci sequence which when each number in the sequence is drawn as a rectangular width creates a spiral.

Isn’t it amazing to think that math can be observed all around us?

But, sometimes using a recursive formula can be a bit tedious, as we continually must rely on the preceding terms in order to generate the next.

So now, let’s turn our attention to defining sequence explicitly or generally. All this means is that each term in the sequence can be calculated directly, without knowing the previous term’s value.

Example

In this problem,

What we will notice is that patterns start to pop-up as we write out terms of our sequences. And it’s in these patterns that we can discover the properties of recursively defined and explicitly defined sequences.

We want to remind ourselves of some important sequences and summations from Precalculus, such as Arithmetic and Geometric sequences and series, that will help us discover these patterns.

Arithmetic Sequence Formula

Geometric Sequence Formula

Summation sequence formulas

Armed with these summation formulas and techniques, we will begin to generate recursive formulas and closed formulas for other sequences with similar patterns and structures.

Example

So, using our known sequences, let’s find a recursive definition for the sequence 4,9,14,19,24,29,…

Now, using our known summation formulae, let’s find a closed definition for the same sequence of 4,9,14,19,24,29,…

Additionally, we will discover a superb procedure for finding the sum of an Arithmetic and Geometric sequence, using Gauss’s discovery of reverse-add and multiply-shift-subtract, respectively.

Example

Suppose we wanted to find the sum of the following sequence: 1,3,5,7,9,..,39.

First, we need to find the closed formula for this arithmetic sequence. To do this, we need to identify the common difference which is the amount that is being added to each term that will generate the next term in the sequence. The easiest way to find it is to subtract two adjacent terms. So, for our current example, if we subtract any two adjacent terms we will notice that the common difference is 2.

Now, we can use the explicit formula to determine the number of terms that we are summing.

Finally, we apply the reverse and add method to find the sum, where we first list all the terms in one direction, then reverse and list all the terms in the opposite direction. In other words, we will “wrap” the series back onto itself, as MathBitsNotebook nicely states.

Examples Using the Formula for Arithmetic Sequence Recursive

Here are a few example questions:

Example 1:

Write the first four terms of the sequence when: a₁ = – 4 and a_n = a_n−1 + 5

Solution:

In recursive formulas, each term is used to produce the next term. Follow the movement of the terms through the steps given below.

Given:

a₁= – 4

And

a_n = a_n−1 + 5 (each term is 5 more than the term before)

n = 2

a₂= a₂₋₁ + 5

a₂ = -4 +5

a₂ = 1

n = 3

a₃ = a₃₋₁ + 5

a₃ = 1 + 5

a₃ = 6

n = 4

a₄ = a₄₋₁ + 5

a₄ = 6 + 5

a₄ = 11

Answer: -4, 1, 6, 11

Example 2: Find the recursive formula when the sequence 2, 4, 6, 8, 10….

Solution:

Considering this sequence, it can be represented in more than one manner. The given sequence can be represented as either an explicit (general) formula or a recursive formula.

Explicit Formula: a_n = 2n

Recursive Formula: a₁ = 2 and a_n = a_n−1 + 2

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