Question:

If (1+x+x2)n=1+a1x+a2x2++a2nx2n\left(1+x+x^{2}\right)^{n} =1+a_{1}x+a_{2}x^{2} +\cdots+a_{2n}x^{2n}, 2a13a2+(2n+1)a2n2a_{1} -3a_{2} +\cdots-\left(2n+1\right)a_{2n} is equal to

Updated On: Jun 6, 2022
  • n
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The Correct Option is B

Solution and Explanation

Given,
(1+x+x2)n=1+a1x+a2x2++a2nx2n\left(1+x+x^{2}\right)^{n}=1+a_{1} x+a_{2} x^{2}+\ldots+ a_{2 n} x^{2 n}
x(1+x+x2)n=x+a1x2+a2x3++a2nx2n+1 \Rightarrow \, x\left(1+x+x^{2}\right)^{n}=x+a_{1} x^{2}+a_{2} x^{3}+ \ldots +a_{2 n} x^{2 n+1}
On differentiating w.r.t. xx, we get
(1+x+x2)n+xn(1+x+x2)n1(1+2x)(1+ \left.x+x^{2}\right)^{n}+x \cdot n\left(1+x+x^{2}\right)^{n-1}(1+2 x)
=1+2a1x+3a2x2++a2n(2n+1)x2n=1+2 a_{1} x+3 a_{2} x^{2}+\ldots+a_{2 n} \cdot(2 n+1) x^{2 n}
On putting x=1x=-1, we get
(11+1)nn(11+1)n1(12)(1-1+1)^{n}-n(1-1+1)^{n-1}(1-2)
=12a1+3a2++a2n(2n+1)=1-2 a_{1}+3 a_{2}+\ldots+a_{2 n}(2 n+1)
1n(1)=12a1+3a2++a2n(2n+1) \Rightarrow \, 1-n(-1)=1-2 a_{1}+3 a_{2}+\ldots+a_{2 n}(2 n+1)
2a13a2(2n+1)a2n=n \Rightarrow \, 2 a_{1}-3 a_{2} \ldots-(2 n+1) a_{2 n}=-n
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Concepts Used:

Binomial Theorem

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.