Given, (1+x+x2)n=1+a1x+a2x2+…+a2nx2n ⇒x(1+x+x2)n=x+a1x2+a2x3+…+a2nx2n+1 On differentiating w.r.t. x, we get (1+x+x2)n+x⋅n(1+x+x2)n−1(1+2x) =1+2a1x+3a2x2+…+a2n⋅(2n+1)x2n On putting x=−1, we get (1−1+1)n−n(1−1+1)n−1(1−2) =1−2a1+3a2+…+a2n(2n+1) ⇒1−n(−1)=1−2a1+3a2+…+a2n(2n+1) ⇒2a1−3a2…−(2n+1)a2n=−n
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
There are (n+1) terms in the expansion of (x+y)n.
The first and the last terms are xn and yn respectively.
From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.