Question

Given that ${(1+x+x^2)}^n=a_0+a_{1}x+a_2{x}^2+\dots +a_{2n}{x}^{2n},$ find the values of
(i) $a_0+a_1+a_2+\dots .+a_{2n}$
(ii) $a_0-a_1+a_2-a_3\dots ..+a_{2n}$

Answer 1

Given,

${(1+x+x^2)}^n=a_0+a_{1}x+a_2{x}^2+\dots +a_{2n}{x}^{2n},$.......(1).

i) Now putting $x=1$ in (1) we get

$(1+1+1{)}^n=a_0+a_1+a_2+\dots .+a_{2n}$

or, $3^n=a_0+a_1+a_2+\dots .+a_{2n}$.
(ii) Now putting $x=-1$ in (1) we get,

$(1-1+1{)}^n=a_0-a_1+a_2-a_3\dots ..+a_{2n}$

or, $1=a_0-a_1+a_2-a_3\dots ..+a_{2n}$.