Question

Let ${\left(1+3x+2x^2\right)}^6=\sum _{r=0}^{12}{a}_r{x}^r$. Identify which of the following statement(s) is(are) correct?

• A. The coefficient of $x^{12}$ is $2^{12}$.
• B. The coefficient of $x^{11}$ is $9\cdot 2^6$.
• C. $\frac{\sum _{r=0}^6{a}_{2r}}{a_{12}}=\frac{3^6}{2}$
• D. $\frac{\sum _{r=1}^6{a}_{2r-1}}{a_{12}}=2\cdot 3^6$

Answer 1

Answer: (B), (C)

${\left(1+3x+2x^2\right)}^6=\sum _{r=0}^{12}{a}_r{x}^r$
${\left(1+3x+2x^2\right)}^6=a_0+a_{1}x+a_2{x}^2+\dots \dots .+a_{12}{x}^{12}$
Putting $x=1a_0+a_1+a_2+\dots \dots +a_{12}=6^6$
Putting $x=-1a_0-a_1+a_2+\dots \dots .+a_{12}=0$
$a_0+a_2+a_4+\dots \dots +a_{12}=3\cdot 6^5$
$a_1+a_3+a_5+\dots \dots +a_{11}=3\cdot 6^5$
$\text{ coefficients of }{x}^{12}=2^6$
$\text{ coefficients of }{x}^{11}:{}^6{C}_6{\left(3x+2x^2\right)}^6={}^6{C}_6\cdot x^6(3+2x{)}^6$
$={}^6{C}_6\cdot {}^6{C}_5\cdot 3^1\cdot 2^5$
$=18\cdot 2^5=9\cdot 2^6$