Question

Coeff. of $x^4$ in ${\log }_e(1+3x+2x^2)$ is :

• A. $15/4$
• B. $17/4$
• C. $-17/4$
• D. None of these

Answer 1

The correct option is C $-17/4$

$\log (1+3x+2x^2)=\log (1+x)+\log (1+2x)$
$={\sum }_{n=1}^{\infty }(-1{)}^{n-1}\frac{x^n}{n}+{\sum }_{n=1}^{\infty }(-1{)}^{n-1}\frac{(2x{)}^n}{n}$
$={\sum }_{n=1}^{\infty }(-1{)}^{n-1}(\frac{1}{n}+\frac{2^n}{n})x^n$
$\therefore$ coeff. of $x^4$ is $(-1{)}^{4-1}(\frac{1}{4}+\frac{2^4}{4})=-\frac{17}{4}$