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General and Middle Terms

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1.In the binomial expansion for `(a + b)^n`, we observe that the first term is `"^nC_0  a^n`, the second term is `"^nC_1a^(n–1)b`, the third term is `"^nC_2 a^(n–2) b^2`, and so on. Looking at the pattern of the successive terms we can say that the `(r + 1)^(th)`term is `"^nC_ra^(n–r)b^r`. The `(r + 1)^th` term is also called the general term of the expansion `(a + b)^n`. It is denoted by `T_(r+1)` Thus `T_(r+1)` = `"^nC_r a^(n–r)b^r`.

2. Regarding the middle term in the expansion `(a + b)n`, we have
(i) If n is even, then the number of terms in the expansion will be n + 1. Since n is even so n + 1 is odd. Therefore, the middle term is `((n + 1 + 1)/2)^(th)` , i.e., `(n/2+1)^(th)` term.
(ii) If n is odd, then n +1 is even, so there will be two middle terms in the expansion, namely `((n+1)/2)^(th)` term and `((n+1)/2 + 1)` term. 
(ii) If n is odd, then n +1 is even, so there will be two middle terms in the expansion, namely `((n+1)/2)^(th)` term and `((n+1)/2 + 1)` term. 

3. In the expansion of `(x +1/x)^(2n)` ,  where x  ≠ 0, the middle term is `((2n +1 +1)/2)^(th)`, i.e., `(n + 1)^(th)` term, as 2n is even.

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