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I have a question from my Mathematics homework which I can't seem to answer.

Write down and simplify the general term in the expression $\left(\frac{2}{x} + \frac{1}{4x^2}\right)^{10}$. Hence, or otherwise, obtain the term in $\frac{1}{x^{13}} $.

Is there any quicker way of getting the answer other than expanding it entirely using Binomial Expansion $\frac{240}{x^{13}}$

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Instead of applying binomial expansion to the expression as written, you can instead rewrite the expression in parentheses as $\frac{8x+1}{4x^2}$, so you have $$\frac{(8x+1)^{10}}{4^{10} x^{20}}.$$ From here, you just need to find the coefficient of $x^{20-13}$ in $(8x+1)^{10}$ and divide by $4^{10}$.

$\binom{10}{7}8^7 / 4^{10} = 120 * 2 = 240$.

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  • $\begingroup$ Thank you, never thought of doing it that way. $\endgroup$
    – Luke Holland
    Commented Nov 5, 2017 at 22:12

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