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Question: If 10th is the only middle term in the expansion of \({(1 + x)^n}\) \(n \in N\). Write its last term...

If 10th is the only middle term in the expansion of (1+x)n{(1 + x)^n} n∈Nn \in N. Write its last term.

Explanation

Solution

Hint: We will start solving this question by using the formula to find the middle term of the given expansion. Then with the help of this formula, we will calculate the last term of the given expansion.

Complete step-by-step answer:
Now, the middle term of an expansion (a+b)n{(a + b)^n} is calculated as below:
If the index n of the expansion is even, then the total number of terms after expansion = n + 1.
So, the middle term is the (n+1+12)(\dfrac{{n + 1 + 1}}{2})th or (n2+1)(\dfrac{n}{2} + 1)th term.
If the index n is odd, so total number of terms after expansion = n + 1.
So, the middle terms of the given expansion are the (n+12)(\dfrac{{n + 1}}{2})th and (n+12+1)(\dfrac{{n + 1}}{2} + 1)th term.
So, in case of n, being odd there are two middle terms, while when n is even, we get only one middle term.
Now, according to the question, there is only one middle term of the expansion (1+x)n{(1 + x)^n}. So, we know that the index n is even. Now, 10th is the middle term. So, from the above property, we get
n2+1=10\dfrac{n}{2} + 1 = 10
On solving, we get n = 18. So, the index of n has the value 18 and the expansion becomes (1+x)18{(1 + x)^{18}}.
Now, from Binomial theorem, we know that the general term of the expansion (a+b)n{(a + b)^n} is written as
Tr+1=nCran−rbr{T_{r + 1}} = {}^n{C_r}{a^{n - r}}{b^r}
So, for the expansion (1+x)18{(1 + x)^{18}}, we get Tr+1=18Cr(1)18−rxr{T_{r + 1}} = {}^{18}{C_r}{(1)^{18 - r}}{x^r}. Now, there are a total of 19 terms after expansion. So,
Last term = T19=T18+1=18C18(1)18−18x18{T_{19}} = {T_{18 + 1}} = {}^{18}{C_{18}}{(1)^{18 - 18}}{x^{18}}
As, nCn=1{}^n{C_n} = 1.
Therefore, last term = x18{x^{18}}.

Note: Whenever we come up with such types of questions, then we will use the formula of the middle term and general term of the expansion. We will follow a few steps to solve this question. First, we will use the relation given in the question. Like in this question, it is given that there is only one middle term. So, from the binomial theorem, we know that the index is even. Then we will use the formula of the middle term to find the value of the index. After finding the value of the index, we will use the formula of the general term to find the term asked in question, like in this question, the last term of the expansion is asked.