Question

The value of $5x-\frac{13}{2}{x}^2+\frac{35}{3}{x}^3-\frac{97}{4}{x}^4+\dots \dots ..$ is

• A. $\log (1+4x+3x^2)$
• B. $1+4x+3x^2$
• C. $1+5x+6x^2$
• D. $\log (1+5x+6x^2)$

Answer 1

The correct option is D $\log (1+5x+6x^2)$

Let $y=5x-\frac{13}{2}{x}^2+\frac{35}{3}{x}^3-\frac{97}{4}{x}^4+\dots \dots ..$

$\Rightarrow y=(2x+3x)-\frac{(2x{)}^2+(3x{)}^2}{2}+\frac{(2x{)}^3+(3x{)}^3}{3}-\frac{(2x{)}^4+(3x{)}^4}{4}+\dots \dots ..$

$=(2x-\frac{(2x{)}^2}{2}+\frac{(2x{)}^3}{3}-\frac{(2x{)}^4}{4}+\dots \dots ..)+(3x-\frac{(3x{)}^2}{2}+\frac{(3x{)}^3}{3}-\frac{(3x{)}^4}{4}+\dots \dots ..)$

We know, $Log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}...$

$\therefore y=\log (1+2x)+\log (1+3x)=\log (1+2x)(1+3x))=\log (1+5x+6x^2)$