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Maths-

General

Easy

Question

$I f blank open vertical bar x close vertical bar less than fraction numerator 1 over denominator 2 end fraction comma blank t h e n blank t h e blank c o e f f i c i e n t blank o f blank x to the power of r end exponent blank i n blank t h e blank e x p a n s i o n blank o f fraction numerator 1 plus 2 x over denominator left parenthesis 1 minus 2 x right parenthesis to the power of 2 end exponent end fraction comma blank i s$

${r 2}^{r}$
${(2 r - 1) 2}^{r}$
$r 2^{2 r + 1}$
$(2 r + 1) 2^{r}$

The correct answer is: $left parenthesis 2 r plus 1 right parenthesis 2 to the power of r end exponent$

$fraction numerator 1 plus 2 x over denominator open parentheses 1 minus 2 x close parentheses to the power of 2 end exponent end fraction equals open parentheses 1 plus 2 x close parentheses left parenthesis 1 minus 2 x right parenthesis to the power of negative 2 end exponent$
$equals open parentheses 1 plus 2 x close parentheses open parentheses 1 plus fraction numerator 2 over denominator 1 factorial end fraction open parentheses 2 x close parentheses plus fraction numerator 2 bullet 3 over denominator 2 factorial end fraction open parentheses 2 x close parentheses to the power of 2 end exponent plus horizontal ellipsis plus fraction numerator 2 bullet 3 horizontal ellipsis r over denominator open parentheses r minus 1 close parentheses factorial end fraction open parentheses 2 x close parentheses to the power of r end exponent plus fraction numerator 2 bullet 3 bullet 4 horizontal ellipsis open parentheses r plus 1 close parentheses open parentheses 2 x close parentheses to the power of r end exponent over denominator r factorial end fraction close parentheses$
$T h e blank c o e f f i c i e n t blank o f blank x to the power of r end exponent equals 2 fraction numerator r factorial over denominator open parentheses r minus 1 close parentheses factorial end fraction 2 to the power of r minus 1 end exponent plus fraction numerator open parentheses r plus 1 close parentheses factorial over denominator r factorial end fraction 2 to the power of r end exponent$
$equals r 2 to the power of r end exponent plus open parentheses r plus 1 close parentheses r to the power of 2 end exponent$
$equals 2 to the power of r end exponent left parenthesis 2 r plus 1 right parenthesis$

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