In this explainer, we will learn how to write explicit and recursive formulas for geometric sequences to find the value of the term in a geometric sequence and how to find a term’s order given its value.
There are many real-world applications of geometric sequences in sciences, business, personal finance, and health. For instance, physicists use geometric sequences to calculate the amount of radioactive material left after any given number of half-lives of the material. During each half-life, the material decays by .
A sequence is an enumerated collection of numbers (or other objects) that usually follow a pattern. The individual elements in a sequence, for , are called the terms and are labeled by the index , which tells us the position of the given term in the sequence.
Now, let’s recall the definition of a geometric sequence.
Definition: Geometric Sequence
A geometric sequence, also known as a geometric progression, is a sequence of nonzero numbers that has a nonzero constant common ratio between any two consecutive terms: where is the term in the sequence.
The geometric sequence in general can also be depicted as follows:
In order to calculate the common ratio of a given geometric sequence, we can divide any term from the sequence by the term that immediately precedes it (e.g., we could divide the third term by the second term or the second term by the first term in the sequence; either way should yield the same number for a geometric sequence). For example, with the geometric sequence , we can clearly see a common ratio between consecutive terms:
This sequence can be depicted as follows:
As we may note from the definition, the recursive formula for the geometric sequence can be written as
In some cases, we may be given a geometric sequence as a relation in this form, using which we can determine the common ratio. A given geometric sequence may be defined using this relation with a given first term . For the example above, the recursive formula for the geometric sequence with is
Up to this point, we have looked at geometric sequences defined either as a sequence of numbers or as a recursive relation , but we can also write this in a general form to yield an exact formula for the term. If we denote the first term as for simplicity, the general form of a geometric sequence is
The second term of the geometric sequence is calculated by multiplying the first term, , by to obtain . The third term is the second term multiplied by , to give .
In other words, every term is multiplied by the same number, , to produce the next term. We may note that, from the general form of the geometric sequence, we can also write an explicit formula for the term as
Let’s demonstrate how this explicit formula for the term can be used to find a specific term in the sequence.
Example 1: Finding the Value of a Certain Term of a Geometric Sequence given Its General Term
Find the value of the second term of the geometric sequence , where .
Answer
Since we have been given the general formula for the term and need to find the second term, we simply have to substitute into the formula (we do this since the sequence is defined for , so we know that the first term corresponds to ; therefore, must give us the second term). This gives us
As we have just seen, using a given formula for the term to find a specific term in the sequence is a fairly straightforward process, since we just have to substitute in the correct value for . Let’s review how this formula can be obtained, given a geometric sequence.
Recall that we previously had the sequence
We can see that, here, the first term is and the common ratio is . Thus, the explicit formula for this sequence is
Now, let’s consider an example where we have to determine the explicit formula for a geometric sequence defined in terms of a recursive formula.
Example 2: Finding the Explicit formula from the Recursive Formula of a Geometric Sequence
The recursive formula for a geometric sequence is and for . Give an explicit formula for the sequence.
Answer
In this example, we want to determine the explicit formula for a given geometric sequence defined in terms of a recursive formula.
Recall that a sequence is geometric if there is a common ratio, , between any two consecutive terms:
We can also rearrange this recursive formula for the geometric sequence as
Now, note that the given relation, , is in this form, and thus the common ratio is
Also, recall that the explicit formula for a geometric sequence with starting value and common ratio is
Therefore, the explicit formula for the given geometric sequence with starting value and common ratio is
We will now look at a few examples where we determine the explicit formula or general term for geometric sequences defined as a sequence of numbers. In the next example, we will consider an increasing and divergent geometric sequence, where all the terms are positive.
Example 3: Finding the General Term of a Given Sequence
Find a formula for the general term of the geometric sequence .
Answer
In this example, we want to find the formula for the general term of a given geometric sequence.
Recall that a sequence is geometric if there is a common ratio between any two consecutive terms. The explicit formula for a geometric sequence with starting value and common ratio is
The first step is to establish the common ratio of the geometric sequence, which we can find from the ratio of any two consecutive terms. We can use the ratio of the 2nd and 1st terms in the sequence to obtain
Therefore, the explicit formula for the given geometric sequence with starting value and common ratio is
Now, let’s consider an example where we determine the general term of an increasing and convergent geometric sequence where all the terms are negative.
Example 4: Finding the General Term of a Given Geometric Sequence
Find, in terms of , the general term of the geometric sequence .
Answer
In this example, we want to find the formula for the general term of a given geometric sequence.
Recall that a sequence is geometric if there is a common ratio between any two consecutive terms. The explicit formula for a geometric sequence with starting value and common ratio is
The first step is to establish the common ratio of the geometric sequence, which we can find from the ratio of any two consecutive terms. We can use the ratio of the 2nd and 1st terms in the sequence to obtain
Therefore, the explicit formula for the given geometric sequence with starting value and common ratio is
In the next example, we will determine the general term of a decreasing and convergent geometric sequence where all the terms are positive.
Example 5: Finding the General Term of a Given Sequence
Find, in terms of , the general term of the sequence .
Answer
In this example, we want to find the formula for the general term of a given geometric sequence.
Recall that a sequence is geometric if there is a common ratio between any two consecutive terms. The explicit formula for a geometric sequence with starting value and common ratio is
The first step is to establish the common ratio of the geometric sequence, which we can find from the ratio of any two consecutive terms. We can use the ratio of the 2nd and 1st terms in the sequence to obtain
Therefore, the explicit formula for the given geometric sequence with starting value and common ratio is
In general, applying the recursive relation repeatedly, we can show that which allows us to determine the value of the term in the sequence, , from the term, :
Now, let’s consider an example where we have to find the first three terms of a geometric sequence given its common ratio and the value of a certain term in the sequence.
Example 6: Finding the Terms of a Geometric Sequence given Its Common Ratio and the Value of a Certain Term
Find the first three terms of a geometric sequence given and the common ratio is .
Answer
In this example, we want to determine the first three terms of a geometric sequence given a value of a certain term and its common ratio.
Recall that a sequence is geometric if there is a common ratio, , between any two consecutive terms
We can also rearrange this recursive formula for the geometric sequence as
Applying this formula repeatedly, we can show that
This allows us to determine the value of the position in the sequence, , from the value, .
Using this formula with and and substituting the given value and the common ratio, we have the following:
The other two terms can be found by multiplying this term by the common ratio :
We note that we could also obtain this answer by simply dividing the sixth term by the common ratio repeatedly to enumerate the previous terms from to , although we are only interested in the first three terms , , and .
Thus, the first three terms of a geometric sequence with a given value of and its common ratio are
In the next example, we will determine three consecutive numbers of a geometric sequence given a particular value for their sum and product.
Example 7: Finding the Terms of a Geometric Sequence given Their Sum and Product
Find the three consecutive numbers of a geometric sequence, given that the sum of the terms is and the product is 216.
Answer
In this example, we want to find three consecutive numbers of a geometric sequence that satisfy a particular relation for the sum and product of the terms.
Recall that a sequence is geometric if there is a common ratio, , between any two consecutive terms:
We can also rearrange this recursive formula for the geometric sequence as
We want to determine the terms , , and in a geometric sequence that satisfies the conditions
Using the recursive formula, we can rewrite these conditions with and ; in other words, if the first number is , then the other two are and .
Using this, the first equation becomes while the second equation becomes
Thus, the second term in the sequence is . In order to find the other two terms, we need to determine the common ratio from the first equation. The first and last terms can then be found from
If we multiply the first equation by and substitute the second equation, we obtain
Thus, the common ratio is or . For , the first and last terms are while for , the first and last terms are
Therefore, in either case, the three consecutive numbers of the geometric sequence are
As we have seen so far, to determine a specific value of the term in a geometric sequence, we have to substitute the given value of in the explicit formula (i.e., for the 5th term, we substitute ).
But what if we want to do the reverse? We want, for a given value in a sequence, to determine the value of , known as the order of the term, which is the position where the value occurs in the sequence, such that the explicit formula for gives this value.
For , we can do this by using the explicit formula by making the subject:
Taking the logarithm of base of both sides of the equation, we obtain:
This works if the terms in the geometric sequence are all positive with or all negative with since the following ratio is always positive in either case:
For example, consider the geometric sequence
This is a decreasing geometric sequence with common ratio and starting value . The exact formula for this sequence is
In order to determine the order of the value , we want to determine the value of such that or
Thus, using the result above, we have
Thus, the order of this value is 7 (i.e., ), which we can also check by substituting into the exact formula:
Now, let’s consider an example where we have to determine the order of a term of a given value for a geometric sequence defined in terms of an explicit formula.
Example 8: Finding the Order of a Term in a Geometric Sequence given Its Value and General Term
Find the order of the term whose value is 4 374 in the geometric sequence .
Answer
In this example, we want to find the order of a particular value that occurs in a given geometric sequence defined with an exact formula.
Recall that a sequence is geometric if there is a common ratio between any two consecutive terms. The order of a value that occurs in a geometric sequence is the position or value of where we obtain the particular value.
We want to determine the value of with or such that
We can rearrange this formula and solve for to obtain
Thus, the order of the term whose value is 4 374 is (i.e., ).
The number of terms in a geometric sequence is equivalent to the order of the last term in the sequence, which is .
In the next example, we will determine the number of terms of a given geometric sequence defined as a sequence of numbers. We will do this by first finding the general term of the sequence and then determining the order of the last term to give us the total number of terms.
Example 9: Finding the Number of Terms of a Given Geometric Sequence
Find the number of terms of the geometric sequence .
Answer
In this example, we want to determine the number of terms of a given geometric sequence.
Recall that a sequence is geometric if there is a common ratio between any two consecutive terms. The order of a value that occurs in a geometric sequence is the position or value of where we obtain the particular value.
The number of terms is equivalent to the order of the last term in the sequence, which we can determine from the explicit formula. The explicit formula for a geometric sequence with starting value and common ratio is
The first step is to establish the common ratio of the geometric sequence, which we can find from the ratio of any two consecutive terms. We can use the ratio of the 2nd and 1st terms in the sequence to obtain
Therefore, the explicit formula for the given geometric sequence with starting value and common ratio is
To find the number of terms, we want to determine the value of with or such that
Now, we take of both sides:
Thus, the number of terms in the given geometric sequence is .
For , we have to take more care as we cannot have a logarithm with a negative base. In this case, the terms in a geometric sequence will alternate signs. If , then all the terms of odd order will have a positive sign and the terms of even order will have a negative sign, while the opposite is true for . Thus, to determine the order of a given value, we must also consider the sign of the starting value and the given value.
In either case, for we have , since is the magnitude of . If we substitute this into the explicit formula, we have
If the starting value and the given value have the same sign, then we have and is an odd number; thus, , and we have
While, if the starting value and given value have opposite signs, then and is an even number; thus, , and we have:
In either case, we can replace the right-hand side by the modulus of the ratio:
We can make the subject by taking the logarithm of base of both sides of the equation:
In fact, this formula is true for any common ratio and starting value .
For example, consider the geometric sequence
This is an alternating geometric sequence with common ratio and starting value . The explicit formula for this sequence is
In this geometric sequence, all the odd-ordered terms are positive, while all the even-ordered terms are negative. Suppose we want to determine the order of the term in this sequence, (i.e., we want to solve the equation ).
Since the value of the term is negative and the first term is positive, we expect the order of the term, , to be an even number, and thus, we have and the equation becomes
We can now rearrange this formula and make the subject as before to obtain the order:
In the final example, we will determine the order of a term with a given value in an alternating geometric sequence defined as a sequence of numbers. We will do this by first finding the general term of the sequence and then determining the order.
Example 10: Finding the Order of a Term in a Given Geometric Sequence given Its Value
Find the order of the term in the geometric sequence .
Answer
In this example, we want to determine the order of a term in a given geometric sequence.
Recall that a sequence is geometric if there is a common ratio between any two consecutive terms. The order of a value that occurs in a geometric sequence is the position or value of where we obtain the particular value, which we can determine from the explicit formula.
The explicit formula for a geometric sequence with starting value and common ratio is
The first step is to establish the common ratio of the geometric sequence, which we can find from the ratio of any two consecutive terms. We can use the ratio of the 2nd and 1st terms in the sequence to obtain
Therefore, the explicit formula for the given geometric sequence with starting value and common ratio is
We want to determine the value of with or such that
Since the signs of the first term and the given term of the sequence are both positive, we expect the order, , to be an odd number, and thus, we have , and the equation becomes
We can rearrange this formula and solve for to obtain the order:
Taking the logarithm of base 2 of both sides of the equation, we have
Thus, the order of the given term in the geometric sequence is or equivalently .
Let’s summarize the key points that we covered in this explainer.
Key Points
- A geometric sequence is a sequence of nonzero numbers defined by a nonzero common ratio between any two consecutive terms
- This formula can also be used to find the next terms in a geometric sequence from the common ratio using the recursive relation: (i.e., each term in a geometric sequence is found by multiplying the previous term by the common ratio).
- In general, applying the recursive relation repeatedly, we can show that which allows us to determine the value of the position in the sequence, , from the value.
- If we denote the starting value as for simplicity, the general form of a geometric sequence is From this general form, we also have an explicit formula for any term in the sequence: We can determine the explicit formula for a geometric sequence by identifying the first term and the common ratio.
- A given geometric sequence may be defined in terms of a set of numbers , the recursive formula with a given first term, or an explicit formula.
- For a given value in a sequence , to determine the value of , the position of the term in the sequence known as the order of the term, we have the general formula