Lesson Explainer: Recursive Formula Of A Sequence - Nagwa

In this explainer, we will learn how to find the recursive formula of a sequence.

Recall that a sequence is just a list of numbers. Some common types of sequence that we meet at this level include arithmetic sequences, where the difference between consecutive terms is constant, and geometric sequences, where there is a common ratio between consecutive terms. Here, we will discuss a powerful method for representing a variety of different sequences that involves knowing the rule to get from one term to the next in a sequence. This means we can then describe the sequence in terms of a recursive formula.

Definition: Recursive Formula of a Sequence

A recursive formula (sometimes called a recurrence relation) is a formula that defines each term of a sequence using a preceding term or terms.

A recursive formula of the form 𝑇=𝑓(𝑇) defines each term of a sequence as a function of the previous term. To generate a sequence from its recursive formula, we need to know the first term in the sequence, 𝑇.

If we know the first term, 𝑇, and the recursive formula 𝑇=𝑓(𝑇), we can use the formula with 𝑛=1 to derive the value of 𝑇 from 𝑇. Once we know the value of 𝑇, we can use the formula with 𝑛=2 to derive the value of 𝑇 from 𝑇, and so on. In this way, we can build up the sequence until it has as many terms as we wish.

This process is best illustrated through some specific examples. Suppose we are asked to find the first four terms of the sequence defined by the recursive formula 𝑇=𝑇+1,𝑇=10.

We already know the first term, which is 𝑇=10.

To find the second term, 𝑇, we substitute 𝑛=1 into the recursive formula 𝑇=𝑇+1 and use the fact that 𝑇=10 to get 𝑇=𝑇+1=10+1=11.

Similarly, to find 𝑇, we substitute 𝑛=2 into the formula and use the fact that 𝑇=11 to get 𝑇=𝑇+1=11+1=12.

Finally, to find 𝑇, we substitute 𝑛=3 into the formula and use the fact that 𝑇=12 to get 𝑇=𝑇+1=12+1=13.

Therefore, the first four terms of this sequence are (10,11,12,13). Note that this is actually an arithmetic sequence with a first term of 10 and a common difference of 1.

As another example, suppose we are asked to find the first five terms of the sequence defined by the recursive formula 𝑇=𝑇−4,𝑇=8.

Clearly, the first term is 𝑇=8.

To find the second term, 𝑇, we substitute 𝑛=1 into the recursive formula 𝑇=𝑇−4 and use the fact that 𝑇=8 to get 𝑇=𝑇−4=8−4=4.

Similarly, to find 𝑇, we substitute 𝑛=2 into the formula and use the fact that 𝑇=4 to get 𝑇=𝑇−4=4−4=0.

To find 𝑇, we substitute 𝑛=3 into the formula and use the fact that 𝑇=0 to get 𝑇=𝑇−4=0−4=−4.

Finally, to find 𝑇, we substitute 𝑛=4 into the formula and use the fact that 𝑇=−4 to get 𝑇=𝑇−4=−4−4=−8.

Therefore, the first five terms of this sequence are (8,4,0,−4,−8). Note that this is actually an arithmetic sequence with a first term of 8 and a common difference of −4. The strength of recursive formulas is that they enable us to describe many different types of sequence, including arithmetic sequences, geometric sequences, and others that we will meet later on in this explainer.

Let us now try an example to practice this skill.

Example 1: Finding the First Five Terms of a Sequence Using Its Recursive Formula

Find the first five terms of the sequence with general term 𝑇=𝑇+5, where 𝑛≥1 and 𝑇=−13.

Answer

Recall that a recursive formula of the form 𝑇=𝑓(𝑇) defines each term of a sequence as a function of the previous term. To generate a sequence from its recursive formula, we need to know the first term in the sequence, that is, 𝑇.

Here, we are given the first term 𝑇=−13 together with the recursive formula 𝑇=𝑇+5.

To find the second term, 𝑇, we substitute 𝑛=1 into the recursive formula and use the fact that 𝑇=−13 to get 𝑇=𝑇+5=−13+5=−8.

Similarly, by substituting 𝑛=2, 3, and 4 into the given recursive formula, we get 𝑇=𝑇+5=−8+5=−3,𝑇=𝑇+5=−3+5=2,𝑇=𝑇+5=2+5=7.

We deduce that the first five terms of this sequence are (−13,−8,−3,2,7). Note that this is actually an arithmetic sequence with a first term of −13 and a common difference of 5.

In some questions, we are given the first few terms of a recursive sequence and must work backward from those terms to find a recursive formula. For example, suppose we are asked to find a recursive formula for this sequence: 3,1,−1,−3,−5,….

We start by inspecting the sequence to see if we can spot the pattern of relationships between the terms. A sensible approach is to check first if each term can be obtained from the previous one by adding or subtracting a constant.

In this case, the terms are decreasing steadily, so we have to subtract. To get from the first term 𝑇=3 to the second term 𝑇=1, we subtract 2. Similarly, to get from the second term 𝑇=1 to the third term 𝑇=−1, we subtract 2, and so on.

Therefore, a recursive formula for this sequence is 𝑇=𝑇−2. To enable someone else to generate this sequence, we also need to state that 𝑇=3.

Now, let us try another example. Suppose we are asked to find a recursive formula for this sequence: 2,10,50,250,1250,….

Again, we start by inspecting the sequence to see if we can spot the pattern of relationships between the terms. In this case, as the terms are increasing, we first check if we can get from one term to the next by adding a constant. To get from the first term 𝑇=2 to the second term 𝑇=10, we add 8. However, to get from the second term 𝑇=10 to the third term 𝑇=50, we add 40, to get from the third term 𝑇=50 to the fourth term 𝑇=250, we add 200, and to get from the fourth term 𝑇=250 to the fifth term 𝑇=1250, we add 1‎ ‎000.

This clearly shows that any recursive formula for the sequence cannot involve adding a constant, because the terms increase by ever larger amounts. Such a pattern suggests we should instead check if we can get from one term to the next by multiplying by a constant. To get from the first term 𝑇=2 to the second term 𝑇=10, we multiply by 5. Similarly, to get from the second term 𝑇=10 to the third term 𝑇=50, we multiply by 5, and so on.

Therefore, a recursive formula for this sequence is 𝑇=5𝑇. To enable someone else to generate this sequence, we also need to state that 𝑇=2.

Let us now try to apply this methodology in the next example.

Example 2: Finding the Recursive Formula of a Sequence given the First Seven Terms

Find a recursive formula for the sequence 486,162,54,18,6,2,23.

Answer

Recall that a recursive formula of the form 𝑇=𝑓(𝑇) defines each term of a sequence as a function of the previous term. To generate a sequence from its recursive formula, we need to know the first term in the sequence, that is, 𝑇.

We start by inspecting the sequence to see if we can spot the pattern of relationships between the terms. First, we check if each term can be obtained from the previous one by adding or subtracting a constant. Here, the terms are decreasing, so to get from one term to the next, we need to subtract.

However, the above diagram clearly shows that any recursive formula for the sequence cannot involve subtracting a constant, because the terms decrease by ever smaller amounts. Such a pattern suggests that we should instead check if we can get from one term to the next by multiplying by a constant.

To find the value of this constant, we can use the following idea.

In general, to get from the first term 𝑇 to the second term 𝑇, we must multiply by 𝑇𝑇, because 𝑇×𝑇𝑇=𝑇.

Similarly, to get from the second term 𝑇 to the third term 𝑇, we must multiply by 𝑇𝑇, because 𝑇×𝑇𝑇=𝑇.

Continuing in this way, we can check the multiplier values for each pair of consecutive terms in the sequence and see if they agree. Notice that as the sequence is decreasing, we will expect any multiplier, which is a number of the form 𝑇𝑇 with 𝑇𝑇 for all natural numbers 𝑛.

A sequence is decreasing if 𝑇𝑇 for all natural numbers 𝑛. A sequence is decreasing if 𝑇𝑇 for all natural numbers 𝑛, so we conclude that it is increasing.

In fact, we could have immediately reached the same conclusion simply by inspecting the recursive formula of the sequence. As 𝑇=𝑇+6, the value of each term is 6 more than the previous one. Thus, 𝑇>𝑇, so the sequence must be increasing.

In general, for any positive number 𝑚, sequences with recursive formulas of the form 𝑇=𝑇+𝑚 are increasing, while those with recursive formulas of the form 𝑇=𝑇−𝑚 are decreasing. It is often more complicated to spot periodic sequences from their recursive formulas.

Our final example tests the same process of recognition.

Example 5: Identifying Whether a Sequence Is Increasing, Decreasing, or Periodic Using Its Recursive Formula

A sequence is given by the recursive formula 𝑇=−𝑇, 𝑇=8. Is this sequence increasing, decreasing, or periodic? If it is periodic, state its order.

Answer

Recall that a recursive formula of the form 𝑇=𝑓(𝑇) defines each term of a sequence as a function of the previous term. To generate a sequence from its recursive formula, we need to know the first term in the sequence, that is, 𝑇.

Here, we are given the first term 𝑇=8 together with the recursive formula 𝑇=−𝑇.

To find the second term, 𝑇, we substitute 𝑛=1 into the formula and use the fact that 𝑇=8 to get 𝑇=−𝑇=−8.

Then, to find the third term, 𝑇, we substitute 𝑛=2 into the formula and use the fact that 𝑇=−8 to get 𝑇=−𝑇=−(−8)=8.

Similarly, by substituting 𝑛=3, 4, and 5 into the formula, we get 𝑇=−𝑇=−8,𝑇=−𝑇=−(−8)=8,𝑇=−𝑇=−8.

Recall that a sequence is increasing if 𝑇>𝑇 for all natural numbers 𝑛. A sequence is decreasing if 𝑇𝑇 for all natural numbers 𝑛.