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1) For how many three-digit positive integers  does the expansion of  \(\left(x^4+\dfrac{1}{x^3}\right)^n\)have a nonzero constant term?

2) What is the sum of the coefficients of the expansion of \(\left(\dfrac{5a}{3}-\dfrac{2b}{3}\right)^{10}\)?

 Aug 24, 2023
 #1
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1) By the Binomial Theorem, the constant term in the expansion of (x4+x31​)n is $(0n​)x4n+(3n​)(x31​)3=x4n+(3n​)x−9.$This term will be nonzero if and only if 4n−9 is even, which is the same as n being even. Therefore, there are 50​ three-digit positive integers n such that the expansion of (x4+x31​)n has a nonzero constant term.

 Aug 24, 2023
 #2
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2) By the Binomial Theorem, the expansion of (5a/3−2b/3)10 is \begin{align*} &\binom{10}{0} \left( \frac{5a}{3} \right)^{10} \left( -\frac{2b}{3} \right)^{0} + \binom{10}{1} \left( \frac{5a}{3} \right)^{9} \left( -\frac{2b}{3} \right)^{1} + \binom{10}{2} \left( \frac{5a}{3} \right)^{8} \left( -\frac{2b}{3} \right)^{2} + \dots + \binom{10}{10} \left( -\frac{2b}{3} \right)^{10} \ &= \frac{5^{10}a^{10}}{3^{10}} - \frac{2^{10}b^{10}}{3^{10}} + \dots + \frac{2^{10}b^{10}}{3^{10}} \ &= \frac{5^{10}a^{10} - 2^{10}b^{10}}{3^{10}}. \end{align*}The sum of the coefficients is then [\frac{5^{10}a^{10} + 2^{10}b^{10}}{3^{10}} = \frac{1}{3^{10}} (5^{10} + 2^{10}) = \frac{9766649}{59049}.]

 Aug 24, 2023

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