Amy has a master's degree in secondary education and has taught math at a public charter high school.
When dealing with sequences in math, both algebraic and geometric, we come across recursive rules. Watch this video lesson to learn how recursion works and how you can use a recursive rule to get to your next number using a previous number.
What is a Recursive Rule?
A recursive rule is like a chicken. You can't get a chicken without an egg. You need the egg to hatch before you can watch a chicken grow up. We define a recursive rule as a rule that continually takes a previous number and changes it to get to a next number. We see recursive rules at work in both arithmetic and geometric sequences. One of the most famous arithmetic sequences of all time is our counting numbers. Think about the numbers you use to count, for a bit. We always start counting with 1. Then we go to 2, then 3, and so on. Each time, we add 1 to the previous number. So our recursive rule here is to add 1 to the previous number with our first number being 1. Just like we can't have a chicken before an egg, we can't count to 2 without the 1. We need the 1 so that we can add the 1 to it to get to 2. Likewise, we need the 2 so we can add a 1 to it to get to the 3. Also, just like we need the egg to get to the chicken, when dealing with recursive rules, we are given what our beginning is. So, with our counting numbers, we are told that our first number is 1.
A Recursive Formula
Sticking to our counting numbers example, our recursive formula here is: a sub n is equal to a sub n minus 1 plus 1 where a sub n minus 1 stands for the previous term and n stands for the position of the current term in our sequence. We are told that our first term, a sub 1, is equal to 1. So, a sub 4 stands for the fourth number in our sequence where n equals 4.
What makes this formula recursive is the a sub n minus 1 part, which tells you that you need to plug in the previous term to find the next. For example, to find the fourth term in the sequence, we need to plug in the value of the third term, a sub 3. Likewise, to find the fifth term, we need to plug in the value of the fourth term, a sub 4.
Using a Recursive Formula
Let's see how to use such a recursive formula, now. Our counting numbers formula tells us that our very first number, a sub 1 is 1. So to find our next number, we use a sub 2 is equal to a sub 1 plus 1. We know that a sub 1 is equal to 1, so we can plug that in to our formula. We get a sub 2 is equal to 1 plus 1 or a sub 2 is equal to 2. With recursive formulas, we have to go term by term. We can't skip a term. For example, we can't find the sixth term without knowing what the fifth term is. Since we've found the first and second term, we can keep going to find our third term, a sub 3. A sub 3 is equal to a sub 2 plus 1. We already know that a sub 2 is equal to 2 so we can plug that in. We get a sub 3 is equal to 2 plus 1, so a sub 3 is equal to 3. We can continue on to the fourth term now. After the fourth term, we can go ahead and find the fifth term. We keep going like this until we reach our desired number in our sequence.
Does this all make sense so far? Let's look at another example to really get a good understanding of recursive rules and formulas.
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This formula tells us that our recursive rule is to add the two previous terms together to get to our term. To get to our third term, we need to add the first and second terms together. We are given what our first and second terms are, so we don't need to calculate those. We only need to calculate for the third term and beyond, so for a sub 3 and so on. According to the formula, to find a sub 3 we need to know the values of a sub 2 and a sub 1 so that we can plug them into the formula. We already know what they are, so now we can figure out a sub 3. When we plug them in we see that a sub 3 is equal to 1 plus 1 which is 2. Our fourth term, a sub 4, is then a sub 3 plus a sub 2. I now know that a sub 3 is 2 and a sub 2 is 1, so plugging these in I get a sub 4 is equal to 2 plus 1 or 3. Our next number, a sub 5, is then a sub 4 plus a sub 3. Plugging in the values that we've found previously, we have a sub 5 is equal to 3 plus 2 or 5.
Wasn't this an interesting lesson? What have we learned? We learned that a recursive rule is a rule that continually takes a previous number and changes it to get to a next number. For example, our counting numbers is a recursive rule because every number is the previous number plus 1. The formula for our counting numbers is a sub n is equal to a sub n minus 1 plus 1 where a sub n minus 1 stands for the previous term and n stands for the position of the current term in our sequence and a sub 1 is 1. The rule here is that every number is equal to the previous number plus 1. What makes any formula a recursive formula is that it uses previous numbers. One thing you need to keep in mind about recursive formulas is that you can't skip around when calculating your numbers. You can't find the eighth term without finding out the seventh term. Likewise, you can't know the seventh term without knowing the sixth term. To use a recursive formula, you plug in your n and you figure out which previous numbers you need and then you plug those in to find your current value.
Once you are done with this lesson you should be able to:
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