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 this proves extremely useful for more advanced topics in mathematics. We begin by recalling the statement of the binomial theorem.
Binomial Theorem
For an integer , where
Sometimes the following notations are used in place of : , , , , and .
In addition to using the general theorem, we are sometimes interested in considering a particular term in the expansion. For this, we use the formula for the general term presented below.
General Term of the Binomial Expansion
In the expansion of , the general term is given by
The general term is variously notated or . The important thing to note when referring to terms by order is that the first term is the term for which . Hence,
This explainer will focus on using the general term to solve problems where we are interested in particular terms in a binomial expansion. In many of these questions, we can always resort to fully expanding the binomial. However, this is often laborious, and using the general term leads to simpler solutions which are less prone to error.
In general, we need to be careful 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 and not to reinforce this fact. Even though we could write the terms of a binomial expansion in any order, there is a standard order which is assumed in most questions which ask for the second, third, or tenth term. The standard order for the terms of the expansion of is decreasing powers of and increasing powers of .
Example 1: Finding the ๐th Term in a Binomial Expansion
Find the third term in the expansion of .
Answer
When presented with a question like this, it would be perfectly legitimate to write the full expansion and then take the coefficient from the appropriate 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
However, we need to remember that the first term starts from . Hence, the third term will be given by and not, as we might naรฏvely think, by . Hence, by setting , , , and , we have
Notice that by using the general term, we can often simplify the calculations we need to make.
Example 2: Finding a Given Term in a Binomial Expansion
Find in the expansion of .
Answer
For the expansion of , the general term is defined as follows:
Hence, by setting , , , and , we have
Since , we can rewrite this as
In the following example, we will see how, by using the general term, we are able to solve for unknowns.
Example 3: 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 .
Answer
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 told that . Hence, we can equate these two expressions for and solve for as follows:
Cancelling the common factors, we have
Hence, .
Example 4: Using the General Term
If the coefficient of the third term in the expansion of is , determine the middle term in the expansion.
Answer
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 given by
Setting , , and , we have
Using the definition of , we can rewrite this as
Since we know the coefficient of this term is , we can write
Multiplying both sides of the equation by 32, we have
Hence, we can find by solving the quadratic equation
We can solve this by factoring or using the quadratic formula to find
Hence, or . Since is a positive integer, we can discard the negative solutions 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 as follows:
Example 5: Finding the Ratio between Consecutive Terms
Consider the expansion of . Find the ratio between the eighth and the seventh terms.
Answer
Recall that the formula for the general term of the binomial expansion of is
Therefore, we can write the general term of the expansion of by setting , , and as follows:
Therefore, the ratio between the eighth and seventh term is given by
Using the rules of exponents, we can simplify this to
Recall that the ratio of consecutive combinations is given by
Hence,
Substituting this into the equation above, we have
In the last 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
We can use this formula to help us solve problems involving the ratios of consecutive terms in binomial expansions.
Example 6: Using Ratios between Consecutive Terms to Solve for Unknowns
Consider the expansion of . Find the values of , , and given that , , and .
Answer
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
Hence, we have
By considering the ratio of these two terms, we can eliminate and and be left with an equation in terms of . Hence,
Simplifying this, we get
Substituting in the values of , , and , we have
Dividing both sides by gives
Cross multiplying by and 6, we have
Hence,
By substituting our value of into equation (1), we have
Using the values of and , we find
Hence, . We can now use the formula for the general term to find the value of and as follows. Using the term , we have
Since , we can rewrite this as which simplifies to
Hence, . Therefore, the final answer is and .
Example 7: 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.
Answer
The question states that when , one of the terms in the expansion is equal to twice its following term. We can write this algebraically as
Hence,
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 through by gives
Gathering like terms, we have
Hence, . Therefore, the two terms that satisfy the given condition are and .
Example 8: Ratios of Nonconsecutive Terms
Consider the expansion of . 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 .
Answer
We would now like to derive an algebraic expression for the ratio of the two terms and . We can do this using the formula for the general term of a binomial expansion. For the expansion of , we can write the general term as
Therefore, we can write the ratio of the two terms and as follows:
Using the powers of exponents, we can rewrite this as
Therefore, the ratios of the coefficients of these terms, which we will denote , is given by
Using the definition of , we can further simplify this expression as follows:
Using the properties of factorials, we have
In the question, we are given two facts that we can express algebraically as and
To solve for , we can take the ratio of to ; this will result in eliminating from the equation, and we will simply be left with an equation in which we will be able to solve:
Using the values of and from the question, we have
Equating these two equations, we have
Multiplying by and dividing by yield
Multiplying by 87, and expanding the parentheses, we have
Gathering all the terms on the left-hand side, we have
We can solve this equation using the quadratic formula to find solutions of and . Since must be a positive integer, we can discard the fractional solution and conclude that . Finally, substituting into the formula for , we can solve for :
Since , we have
Hence, which gives .
Key Points
- Using the general term for the binomial expansion often simplifies calculations in which we are only interested in specific terms.
- Consecutive terms in a binomial expansion are related by the formula The relationship between consecutive terms can be useful if we only require the first few terms of an expansion.