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# Recursive formulas for arithmetic sequences

Learn how to find recursive formulas for arithmetic sequences. For example, find the recursive formula of 3, 5, 7,...
Before taking this lesson, make sure you are familiar with the basics of arithmetic sequence formulas.

## 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, comma, 5, comma, 7, comma, point, point, point along with the interpretation for each part.
$\begin{cases}a(1) = 3&\leftarrow\gray{\text{the first term is 3}}\\\\ a(n) = a(n-1)+2&\leftarrow\gray{\text{add 2 to the previous term}} \end{cases}$
In the formula, n is any term number and a, left parenthesis, n, right parenthesis is the n, start superscript, start text, t, h, end text, end superscript term. This means a, left parenthesis, 1, right parenthesis is the first term, and a, left parenthesis, n, minus, 1, right parenthesis is the term before the n, start superscript, start text, t, h, end text, end superscript term.
In order to find the fifth term, for example, we need to extend the sequence term by term:
a, left parenthesis, n, right parenthesisequals, a, left parenthesis, n, minus, 1, right parenthesis, plus, 2
a, left parenthesis, 1, right parenthesisequals, start color #0d923f, 3, end color #0d923f
a, left parenthesis, 2, right parenthesisequals, a, left parenthesis, 1, right parenthesis, plus, 2equals, start color #0d923f, 3, end color #0d923f, plus, 2equals, start color #aa87ff, 5, end color #aa87ff
a, left parenthesis, 3, right parenthesisequals, a, left parenthesis, 2, right parenthesis, plus, 2equals, start color #aa87ff, 5, end color #aa87ff, plus, 2equals, start color #11accd, 7, end color #11accd
a, left parenthesis, 4, right parenthesisequals, a, left parenthesis, 3, right parenthesis, plus, 2equals, start color #11accd, 7, end color #11accd, plus, 2equals, start color #e07d10, 9, end color #e07d10
a, left parenthesis, 5, right parenthesisequals, a, left parenthesis, 4, right parenthesis, plus, 2equals, start color #e07d10, 9, end color #e07d10, plus, 2equals, 11
Cool! This formula gives us the same sequence as described by 3, comma, 5, comma, 7, comma, point, point, point

1) Find b, left parenthesis, 4, right parenthesis in the sequence given by $\begin{cases}b(1)=-5\\\\ b(n)=b(n-1)+9 \end{cases}$
b, left parenthesis, 4, right parenthesis, equals

## Writing recursive formulas

Suppose we wanted to write the recursive formula of the arithmetic sequence 5, comma, 8, comma, 11, comma, point, point, point
The two parts of the formula should give the following information:
• The first term left parenthesiswhich is start color #0d923f, 5, end color #0d923f, right parenthesis
• The rule to get any term from its previous term left parenthesiswhich is "add start color #ed5fa6, 3, end color #ed5fa6"right parenthesis
Therefore, the recursive formula should look as follows:
$\begin{cases}c(1)=\greenE 5\\\\ c(n)=c(n-1)\maroonC{+3} \end{cases}$

2) What is the recursive formula of the sequence 12, comma, 7, comma, 2, comma, point, point, point ?

3) Complete the missing values in the recursive formula of the sequence 2, comma, 8, comma, 14, comma, point, point.
$\begin{cases}e(1)=A\\\\ e(n)=e(n-1)+B \end{cases}$
A, equals
B, equals

4) Complete the missing values in the recursive formula of the sequence minus, 1, comma, minus, 4, comma, minus, 7, comma, point, point, point.
$\begin{cases}f(1)=A\\\\ f(n)=f(n-1)+B \end{cases}$
A, equals
B, equals

### Reflection question

5) Here is the general recursive formula for arithmetic sequences.
$\begin{cases}g(1)=A\\\\ g(n)=g(n-1)+B \end{cases}$
What is the common difference of the sequence?

## Want to join the conversation?

• Do we have to find the term number before the other ones to find a certain term number?
• Yes, when using the recursive form we have to find the value of the previous term before we find the value of the term we want to find. For example, if we want to find the value of term 4 we must find the value of term 3 and 2. We are already given the value of the first term.

In other words to find any term beyond the first term we have to start at the beginning which would be the 2nd term and continue to calculate the value of each proceeding term until we have reached the term we want to find.

Makes sense?
• I don't quite understand the purpose of the recursive formula. I understand how it works, and according to my understanding, in order to find the nth term of a sequence using the recursive definition, you must extend the terms of the sequence one by one. But doesn't this defeat the purpose of it? Isn't the purpose of a formula to find out the nth term of the sequence without computing all the terms before it?

Am I missing something critical here?
• Formulas are just different ways to describe sequences. Each description emphasizes a different aspect of the sequence, which may or may not be useful in different contexts. For example, we may be comparing two arithmetic sequences to see which one grows faster, not really caring about the actual terms of the sequences. In this case, the recursive definition gives the rate of change a little more directly than the standard formula.

There are also sequences that are much easier to describe recursively than with a direct formula. For example, the Fibonacci sequence, which starts {0, 1, 1, 2, 3, 5, 8...}, with each successive term being the sum of the previous two. While this does have a closed formula, it's very complex and unwieldy.
• What good would this stuff do us in the real world? PLZ tell me!
• Sequences are really important in real life, as they play a key part in areas such as statistics, finance and even in controlling the growth of a species!! One example can be you planning for a vacation. You would look at the temperature of your choosen vacation spot for each month and then decide which month is the apt time to visit the place. Invariably, these temperatures are a sequence and are stored in a set. Who would have known that to enjoy your vacation, you would have to brush up on your sequences first!!
• Hi. I don't understand what "common difference" stands for.
• For an arithmetic sequence, we add a number to each term to get the next term. That number is the common difference.
So for {0, 3, 6, 9...}, we're adding 3 each time. So the common difference is 3.

Note: only arithmetic sequences have a common difference.
• How would you solve something like:
f(n)=f(n-1)+f(n-1)-f(n-2)+35
f(1)=5
f(2)=30
f(n)= Some number in the 10thousands, not sure what numbers work in this particular scenario
For n
Is there any way to solve this without going through each and every step?
• Well, lets see what the first few terms are, f(1) = 5, f(2) = 30, f(3) = 30+30-5+35= 90, f(4) = 90 + 90 - 30+35 = 185, f(5) = 185 + 185 - 90 + 35 = 315, f(6) = 315 + 315 - 185 + 35 = 480. So we have a sequence of 5, 30, 90, 185,315, 480 ... We then can find the first difference (linear) which does not converge to a common number (30-5 = 25, 90-30=60, 185-90=95, 315-185=130, 480-315=165. Then the second difference (60 - 25 = 35, 95-60 = 35, 130-95=35, 165-130 = 35) gives a second common difference, so we know that it is quadratic. I do not know any good way to find out what the quadratic might be without doing a quadratic regression in the calculator, in the TI series, this is known as STAT, so plugging the original numbers in, I ended with the equation:
f(x) = 17.5x^2 - 27.5x + 15. This gives us any number we want in the series. Maybe these having two levels of numbers to calculate the current number would imply that it would be some kind of quadratic function just as if I only had 1 level, it would be linear which is easier to calculate by hand.
• if the sequence is 4,8,12,16... and arithmetic how could I write a recessive and explicit formula for that sequence?
• The recursive formula for the arithmetic set{4,8,12,16,...} is: {a(n) = 4 when n = 1
a(n-1) + 4 when n > 1
The explicit formula for the same set is: a(n) = 4 + 4(n-1). I hope this makes sense. Thank you.
• When ever we are doing recursive formulas why do we add that x(n-1)+ something, why do we do that
• That would be the rule to get any term from its previous term

For example,
c(1)=5
in order to find any term, we simply need to put the nth term into
c(n)=c(n−1)+3
​where +3 is the common difference

Only arithmetic sequences have a common difference
The common difference of an A.P. can be positive, negative or zero