Prove that ∑ C . General and Middle Terms . In the binomial expansion for ( a + b ) n , we observe that the first term is n C a n , the second term is n C a n – b , the third term is n C a n – b , and so on. Looking at the pattern of the successive terms we can say that the ( r + ) th term is n C r a n – r b r .
The ( r + ) th term is also called the general term of the expansion ( a + b ) n . It is denoted by T r + . Thus T r + = n C r a n–r b r . .
Regarding the middle term in the expansion ( a + b ) n , we have If n is even, then the number of terms in the expansion will be n + . Since n is even so n + is odd. Therefore, the middle term is th , i.e., th + term. For example, in the expansion of ( x + y ) , the middle term is th + i.e., th term.
If n is odd, then n + is even, so there will be two middle terms in the MATHEMATICS expansion, namely, th term and th n + term. So in the expansion ( x – y ) , the middle terms are th , i.e., th and th , i.e., th term. . In the expansion of + , where x ≠ , the middle term is th , i.e., ( n + ) th term, as n is even.
It is given by n C n x n x = n C n (constant). This term is called the term independent of x or the constant term. Example Find a if the