Let $\mathrm{g}\left( x \right)=\frac{5x-4{{x}^{2}}}{\left( 1+x \right){{\left( 1-2x \right)}^{2}}}$

Question

Let $\mathrm{g}\left( x \right)=\frac{5x-4{{x}^{2}}}{\left( 1+x \right){{\left( 1-2x \right)}^{2}}}$. Express $\mathrm{g}\left( x \right)$ in the form $\frac{A}{1+x}+\frac{B}{{{\left( 1-2x \right)}^{2}}}+\frac{C}{1-2x}$, where $A$, $B$, and $C$ are constants to be determined. [2] The expansion of $\mathrm{g}\left( x \right)$, in ascending powers of $x$, is ${{c}_{0}}+{{c}_{1}}x+{{c}_{2}}{{x}^{2}}+{{c}_{3}}{{x}^{3}}+\ldots +{{c}_{r}}{{x}^{r}}+\ldots$. Find the values of ${{c}_{0}}$, ${{c}_{1}}$, ${{c}_{2}}$, and ${{c}_{3}}$. [3] Express ${{c}_{r}}$ in terms of $r$. [1] Find the range of values of $x$ for which the expansion is valid. [2]

Accepted Answer

(i) $A=-1$, $B=1$, $C=0$ (ii) $c_0=0$, $c_1=5$, $c_2=11$, $c_3=33$ (iii) $c_r=(-1)^{r+1}+(r+1)2^r$ (iv) $-\dfrac12<x<\dfrac12$

2022 HCI CT1 Q1 (b)

2022 HCI CT1 Q1 (b)

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