In this explainer, we will learn how to find a specific term inside a binomial expansion and find the relation between two consecutive terms.
The binomial theorem provides us with a general formula for expanding binomials raised to arbitrarily large powers. Being confident at using the binomial theorem proves extremely useful for more advanced topics in mathematics. We begin by recalling the statement of the binomial theorem.
Theorem: Binomial Theorem
For an integer , where
It is worth noting that if you read more widely into this topic, you may come across alternative notations for , namely , , , and .
In addition to using the general theorem, we can consider a particular term in the expansion. For this, we use the formula for the general term presented below.
Formula: General Term of the Binomial Expansion
In the expansion of , the general term is
The important thing to note here, when referring to terms by their order, is that the first term, , is the term for which .
This explainer will focus on using the general term to solve problems involving particular terms in a binomial expansion. In many of these questions, we can resort to fully expanding the binomial. However, this is often laborious, and using the general term leads to simpler, more concise solutions that are less prone to error.
As previously mentioned, we need to remember that the first term of a binomial expansion is the term for which . It is a common mistake to assume that the first term is when . However, this is incorrect and hence why we tend to define rather than to reinforce this fact. Even though we could write the terms of a binomial expansion in any order, there is a standard order that is assumed in most questions that ask for the second, third, or perhaps tenth term. The standard order for the terms of the expansion of is decreasing powers of and increasing powers of .
Example 1: Finding a Specific Term in a Binomial Expansion
Find the third term in the expansion of .
When presented with a question like this, it would be perfectly legitimate to write the full expansion and then identify the third term. However, appealing to the formula for the general term simplifies our calculation. This is the method we will demonstrate here. Recall that the formula for the general term for the expansion of is
Recall that the first term in the expansion corresponds to the general term with . Hence, the third term will be given by rather than . Hence, by setting , , , and , we have
Therefore, the third term of the expansion is .
Notice that, by using the general term, we can often simplify the calculations we need to make. In our second example, we will look at a very similar concept but with a binomial raised to a higher power.
Example 2: Finding a Given Term in a Binomial Expansion
Find in the expansion of .
For the expansion of , the general term is defined as follows:
Hence, by setting , , , and , we have
Since , we can rewrite this as
We can then simplify this using the laws of exponents as follows:
Therefore, the fourth term of the expansion, , is equal to .
As we have seen in the previous two examples, we can use the general term of a binomial expansion to find a specified term of the expansion, but equally, we could be asked to identify the coefficient of a specified term as we will demonstrate in our next example.
Example 3: Finding the Coefficient of a Specified Term in a Binomial Expansion
Determine the coefficient of in the expansion of .
The first thing to note in this example is that we can rewrite the binomial expression as . We then need to identify any terms that result in an exponent of . Recall that the general term of the expansion of is . We have , , and , so substituting these expressions into the general term leads to
Using the power law, we can write this as
Since we want the term with , we want the exponent of to be equal to . This means which leads to . We can verify this by substituting into the general term:
We can see that the coefficient of the term is exactly . Hence, the coefficient of is 15.
In the following example, we will see how, by using the general term, we are able to solve for unknowns.
Example 4: Using the General Term to Find Unknowns
The terms of the expansion of are arranged according to the descending powers of . Given that , find the value of .
To solve this problem, we can use the formula for the general term of the binomial expansion to find an alternative expression for . We can then equate the two expressions and solve for . Recall that the general term of the binomial expansion of is given by
Setting , , , and , we have
In the question, we are given that . Hence, we can equate these two expressions for as follows:
We can see that both sides of the equation contain the factor , and by equating coefficients, we can write
Taking the cube root of both sides of the equation, we obtain .
In our next example, let us look at how we can use the general term to solve a multistep problem.
Example 5: Using the General Term
If the coefficient of the third term in the expansion of is , determine the middle term in the expansion.
Using the formula for the general term of the binomial expansion, we can find an expression for the coefficient of the third term in terms of . Using this, we can solve for and then find the middle term of the expansion. Recall that the general term of the binomial expansion of is equal to
Since there is a negative sign in the binomial expression, we can begin by writing
Note that the expression for the general term begins at ; therefore, to calculate the third term, we need to set . Substituting , , and , we have
Recalling that , we can rewrite this as which simplifies to
Since we are given that the coefficient of this term is , we can write
Multiplying both sides of the equation by 32, we have
If we subtract 132 from both sides of the equation, this leads to the quadratic equation
We can solve this for by factoring to find
Hence, or . The binomial theorem only applies for the expansion of a binomial raised to a positive integer power. Therefore, must be a positive integer, so we can discard the negative solution and hence . We can now use this to find the middle term of the expansion. Since , there will be thirteen terms in the expansion, and the middle term will be the seventh term. Hence, we can use the formula for the general term to find the seventh term of this expansion. Again, since begins at , the seventh term in the expansion corresponds to . Substituting this value into the formula for the general term, we obtain
Therefore, the middle term in the expansion is .
If we calculate two consecutive terms in a binomial expansion, we can then find the ratio between them. For the terms and , the ratio between them is . We will demonstrate how to calculate this in our next example.
Example 6: Finding the Ratio between Consecutive Terms
Consider the expansion of . Find the ratio between the eighth and the seventh terms.
Recall that the formula for the general term of the binomial expansion of is
Here, represents the term in the binomial expansion. This means that the seventh term, , is obtained using , and the eighth term, , is obtained using . We can write the general term of the expansion of by setting , , and as follows:
As mentioned earlier, we can calculate the seventh term by substituting :
Similarly, we can calculate the eighth term by substituting :
Therefore, the ratio between the eighth and seventh terms is given by
Using the rules of exponents, we can simplify this to
Recall that the ratio of consecutive combinations is given by
Substituting this into the equation above, we have
Therefore, the ratio between the eighth and seventh terms in the binomial expansion is .
In the previous example, we considered the ratio between two consecutive terms. This is in fact a common thing to consider, and there is a simple expression for this in general. Consider the two consecutive terms and of the expansion of ; using the formula for the general term, we can write their ratio as follows:
Using the rules of exponents, we can simplify this to
We can now use the formula for ratios of consecutive combinations, to rewrite this as
Formula: The Ratio between Consecutive Terms of a Binomial Expansion
For two consecutive terms and in the expansion , the ratio between them is
We can use this formula to help us solve problems involving the ratios of consecutive terms in binomial expansions.
Example 7: Using Ratios between Consecutive Terms to Solve for Unknowns
Consider the expansion of , where is positive. Find the values of , , and given that , , and .
One of the simplest ways to address this problem is by considering the ratios of consecutive terms. Recall that the ratio of two consecutive terms and in the expansion of is given by
Substituting and , respectively, along with the values of , , and , we obtain
By considering the ratio of these two ratios, we can eliminate and and be left with an equation in terms of by dividing equation (1) by equation (2). Hence,
This is equivalent to
Simplifying this, we get
Substituting in the values of , , and , we have
Dividing both sides by gives
Cross multiplying by and 6, we have
This can be solved as follows:
By substituting our value of into equation (1), we have
Dividing both sides of the equation by leads to . Multiplying both sides of the equation by a gives us . We can now use the formula for the general term to find the value of and as follows. We can write the general term by substituting and set it equal to the given value:
Since , we can rewrite this as
Recall that and therefore
We have which simplifies to
Taking the 10th root of both sides of the equation leads to . Since we are given that must be positive, we obtain . We know that ; therefore, . Our final answer is and .
Example 8: Using the Ratio of Consecutive Terms
Consider the binomial expansion of in ascending powers of . When , one of the terms in the expansion is equal to twice its following term. Find the position of these two terms.
Recall first that the general term of the expansion of is
This leads to an ascending order of powers of when we substitute and and systematically increase the value of . The question states that when , one of the terms in the expansion, arranged in ascending powers of , is equal to twice its following term. We can write this algebraically as
Recall that for the binomial expansion of , the ratio between consecutive terms is given by
Setting , , and , we can rewrite this as
Since this is equal to a half when , we can write
Multiplying both sides of the equation by gives
Adding to both sides of the equation, we have
Dividing through by 29 gives us . Therefore, the two terms that satisfy the given condition are and .
We can also calculate ratios between nonconsecutive terms using similar methods, though the process is a little more involved. We will demonstrate in our final example.
Example 9: Ratios of Nonconsecutive Terms
Consider the expansion of , where is a positive constant. Determine the values of and , given that the ratio between the coefficients of and is equal to and that the ratio between the coefficients of and is equal to .
One possible approach to this problem would be to directly calculate expressions for the ratios between the coefficients of the twelfth and fourteenth terms and the seventh and ninth terms by applying the formula for consecutive terms. For example, for the ratio between the twelfth and fourteenth terms, we could use the relation
However, given that we need to calculate two ratios and ultimately form two equations, we will start by deriving an algebraic expression for the ratio of the two terms and and then substitute for the necessary values of (and the other variables) to find the particular expressions for the two ratios.
We can do this by starting with the formula for the ratio of two consecutive terms, , which is
We can then form the following equation:
We can calculate the ratio by substituting into the formula for the ratio of consecutive terms:
We have information about the ratio between the coefficients of the twelfth and fourteenth terms and the ratio between the coefficients of the seventh and ninth terms, so we need to start by looking at the reciprocal of our above equation:
In the question, we are told the ratio between the coefficients of the twelfth and fourteenth terms and likewise the ratio between the coefficients of the seventh and ninth terms. By substituting for , , and , we have
Similarly, substituting gives
As we know the value of the ratio of the coefficients, we can remove the variable from the above ratios and form the following equations:
Dividing equation (3) by equation (4), noting that this is the same as multiplying the equation (3) by the reciprocals of each side of the equation (4), we have noting that this is the same as multiplying the first equation by the reciprocals of each side of the second equation, we have
This simplifies to
Multiplying by and dividing by yields
Multiplying by 87, and multiplying through the parentheses, we have
Gathering all the terms on the left-hand side, we have
We can solve this equation using the quadratic formula that we recall is for the quadratic , to find the solutions and . Since must be a positive integer, we can discard the fractional solution and conclude that . Finally, we need to substitute into either equation (3) or (4) to find . We will substitute into equation (4):
Hence, which gives . We are told that is in fact a positive constant that gives us a final answer of .
Let us finish by recapping a few important concepts from this explainer.
- Using the general term for the binomial expansion often simplifies calculations in which we are only interested in specific terms and their coefficients.
- The formula for the general term for the binomial expansion is In particular, we should note that the first term corresponds to . This means that the general term is obtained by using in the general form.
- Consecutive terms in the binomial expansion are related by the formula