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Sum of Binomial Coefficients of Even Numbered Terms

If n is the...

Question

If $n$ is the degree of the polynomial, ${[\frac{2}{\sqrt{5 x^{3} + 1} - \sqrt{5 x^{3} - 1}}]}^{8} + {[\frac{2}{\sqrt{5 x^{3} + 1} + \sqrt{5 x^{3} - 1}}]}^{8}$ and $m$ is the coefficient of $x^{n}$ in it, then the ordered pair $(n, m)$ is equal to

A

(12, (20)^{4})

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B

(8, 5 (10)^{4})

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C

(24, (10)^{8})

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D

(12, 8 (10)^{4})

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Solution

The correct option is A $(12, (20)^{4})$
${[\frac{2}{\sqrt{5 x^{3} + 1} - \sqrt{5 x^{3} - 1}}]}^{8} + {[\frac{2}{\sqrt{5 x^{3} + 1} + \sqrt{5 x^{3} - 1}}]}^{8}$

Rationalise the polynomial,

$2^{8} {[\frac{1}{\sqrt{5 x^{3} + 1} - \sqrt{5 x^{3} - 1}} \times \frac{\sqrt{5 x^{3} + 1} + \sqrt{5 x^{3} - 1}}{\sqrt{5 x^{3} + 1} + \sqrt{5 x^{3} - 1}}]}^{8} + {[\frac{1}{\sqrt{5 x^{3} + 1} + \sqrt{5 x^{3} - 1}} \times \frac{\sqrt{5 x^{3} + 1} - \sqrt{5 x^{3} - 1}}{\sqrt{5 x^{3} + 1} - \sqrt{5 x^{3} - 1}}]}^{8}$

$= 2^{8} {[\frac{\sqrt{5 x^{3} + 1} + \sqrt{5 x^{3} - 1}}{(5 x^{3} + 1) - (5 x^{3} - 1)}]}^{8} + {[\frac{\sqrt{5 x^{3} + 1} - \sqrt{5 x^{3} - 1}}{(5 x^{3} + 1) - (5 x^{3} - 1)}]}^{8}$

$= \frac{2^{8}}{2^{8}} [{[\sqrt{5 x^{3} + 1} + \sqrt{5 x^{3} - 1}]}^{8} + (\sqrt{5 x^{3} + 1} - \sqrt{5 x^{3} - 1})^{8}]$

$= [(a + b)^{8} + (a - b)^{8}]$

we know, $(a + b)^{8} =^{8} C_{0} a^{8} b^{0} +^{8} C_{1} a^{7} b^{1} + . . . +^{8} C_{8} a^{0} b^{8}$

$(a - b)^{8} =^{8} C_{0} a^{8} b^{0} -^{8} C_{1} a^{7} b^{1} + . . . +^{8} C_{8} a^{0} b^{8}$

$(a + b)^{8} + (a - b)^{8} = 2 [^{8} C_{0} a^{8} b^{0} +^{8} C_{2} a^{6} b^{2} +^{8} C_{4} a^{4} b^{4} +^{8} C_{6} a^{2} b^{6} +^{8} C_{8} a^{0} b^{8}]$

Thus, our expression becomes,
$= 2 [^{8} C_{0} (\sqrt{5 x^{3} + 1})^{8} +^{8} C_{2} (\sqrt{5 x^{3} + 1})^{6} (\sqrt{5 x^{3} - 1})^{2} +^{8} C_{4} (\sqrt{5 x^{3} + 1})^{4} (\sqrt{5 x^{3} - 1})^{4} +^{8} C_{6} (\sqrt{5 x^{3} + 1})^{2} (\sqrt{5 x^{3} - 1})^{6} +^{8} C_{8} (\sqrt{5 x^{3} - 1})^{8}]$

$= 2 [^{8} C_{0} (5 x^{3} + 1)^{4} +^{8} C_{2} (5 x^{3} + 1)^{3} (5 x^{3} - 1) +^{8} C_{4} (5 x^{3} + 1)^{2} (5 x^{3} - 1)^{2} +^{8} C_{6} (5 x^{3} + 1) (5 x^{3} - 1)^{3} +^{8} C_{8} (5 x^{3} - 1)^{4}]$

From this, we can clearly see that the degree of the polynomial is $12$ , hence $h = 12$ which means the option (2) & (3) are incorrect.
now, for $m$ , let collect the coefficients of $x^{12}$ from each term.

coefficient of $x^{12} = 2 [^{8} C_{0} 5^{4} +^{8} C_{2} 5^{4} +^{8} C_{4} 5^{4} +^{8} C_{6} 5^{4} +^{8} C_{8} 5^{4}]$

$= 2 [5^{4} \times 2^{7}]$

$= 5^{4} \times 2^{4} \times 2^{4}$

$= 10^{4} \times 2^{4}$

$= 16 (10^{4})$

$= (20)^{4}$

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{[\frac{2}{\sqrt{5 x^{3} + 1} - \sqrt{5 x^{3} - 1}}]}^{8} + {[\frac{2}{\sqrt{5 x^{3} + 1} + \sqrt{5 x^{3} - 1}}]}^{8}

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