# Binomial Theorem, General Term in a Binomial Expansion, Middle Terms in a Binomial Expansion

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## General Term in a Binomial Expansion

Let us examine various terms in the expansion of , i.e., in

• We observe that, the first term is , i.e., ;

• The second term is , i.e., ;

• The third term is , i.e., ; and so on.

From the above, we can generalise that

• The term is , i.e., .

• If we denote this term by , we have ,

• is generally referred to as the general term of the binomial expansion.

• Let us now consider some examples and find the general terms of some expansions.

Example:

Find the term in the expansion of , where is a natural number. Verify your answer for the first term of the expansion.

Solution:

The general term of the expansion is given by:

Hence, the term in the expansion is .

On expanding, we note that the first term is or .

Using, we find the first term by putting .

Since

This verifies that the expression for is correct for .

## Middle Terms in a Binomial Expansion

• Now you are familiar with the general term of an expansion, let us see how we can obtain the middle term (or terms) of a binomial expansion.

• Recall that the number of terms in a binomial expansion is always one more than the exponent of the binomial. This implies that if the exponent is even, the number of terms is odd, and if the exponent is odd, the number of terms is even.

• Thus, while finding the middle term in a binomial expansion, we come across two cases:

Case 1:

When is even. To study such a situation, let us look at a particular value of , say . Then the number of terms in the expansion will be . From Fig., you can see that there are three terms on either side of the fourth term. The Number of Term in Expansionin 7The Number of term in expansionin 7

In general, when the exponent of the binomial is even, there are terms on either side of the term. Therefore, the term is the middle term.

Case 2:

• When is odd, Let us take as an example to see what happens in this case. The number of terms in the expansion will be . Looking at Fig., do you find any one middle term in it? There is not.

• But we can partition the terms into two equal parts by a line as shown in the figure. We call the terms on either side of the partitioning line taken together, the middle terms. This is because there are an equal number of terms on either side of the two, taken together. The Number of Term in Expansionin 8The Number of term in expansionin 8
• Thus, in this case, there are two middle terms, namely, the fourth,

• i.e., and the fifth, i.e., terms

• Similarly, if , then the and the terms, i.e., the and terms are two middle terms, as is evident from Fig.

• From the above, we conclude that The Number of Term in Expansionin 14The Number of term in expansionin 14
• When the exponent of a binomial is an odd natural number, then the and terms are two middle terms in the corresponding binomial expansion.

Example:

Find the middle term in the expansion of .

Solution:

Here (an even number).

Therefore, the , i.e., the term is the middle term.

Putting in the general term

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