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AP EAMCET Engineering 20 Apr 2019 Shift 2 Paper

Section: Mathematics

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Question : 18 of 160

Marks: +1, -0

If

(1+x+x^2)^n

c_0+c_1x+c_2x^2

+ ... , then the value of

c_0c_1-c_1c_2+c_2c_3

- ... is

$(-1)^n$
0
$2^n$
$3^n$

Solution:

Consider the equation.

(1+x+x^{2})^{n}=c_{0}+c_{1}x+c_{2}x^{2}+\cdots

Substitute

\;\;\frac{1}{x}

for

x

in the above equation.

\;\;\frac{(1-x+x^{2})^{n}}{x^{2n}} = \;\;\frac{c_{0} x^{2n} - c_{1} x^{2n-1} + c_{2} x^{2n-2} + c_{3} x^{2n-3} + \cdots}{x^{2n}}

(1-x+x^{2})^{n}=c_{0} x^{2n}-c_{1} x^{2n-1}+c_{2} x^{2n-3}+\cdots

Now

c_{0} x^{2n}+c_{1} x^{2n-1}+c_{2} x^{2n-3}+\cdots = \;\text{ Coeff of }\; x^{2n+1} \;\text{ in }\;(1+x+x^{2})^{n}(1-x+x^{2})^{n}

= \;\text{ Coeff of }\; x^{2n+1} \text{ in }((1+x^{2})^{2}-x^{2})^{n}

= \;\text{ Coeff of }\; x^{2n+1} \text{ in }(1+x^{4}+x^{2})^{n}

=0

since the coefficient of

x^{k}

in the expansion if

(1+x^{4}+x^{2})^{n}

exist if

k

is even other it will be zero and

2n+1

is odd

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