Gunsaz

2022-10-02

How do you write the first five terms of the sequence ${a}_{n}=\frac{10}{{n}^{\frac{2}{3}}}$?

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Colin Dougherty

Expert

We have: ${a}_{n}=\frac{10}{{n}^{\frac{2}{3}}}$
To evaluate the first five terms, simply replace n with the number of the term:
$⇒{a}_{1}=\frac{10}{{1}^{\frac{2}{3}}}=\frac{10}{1}=10$
$⇒{a}_{2}=\frac{10}{{2}^{\frac{2}{3}}}=\frac{10}{{\left({2}^{2}\right)}^{\frac{1}{3}}}=\frac{10}{\sqrt[3]{4}}$
$⇒{a}_{3}=\frac{10}{{3}^{\frac{2}{3}}}=\frac{10}{{\left({3}^{2}\right)}^{\frac{1}{3}}}=\frac{10}{\sqrt[3]{9}}$
$⇒{a}_{4}=\frac{10}{{4}^{\frac{2}{3}}}=\frac{10}{{\left({4}^{2}\right)}^{\frac{1}{3}}}=\frac{10}{\sqrt[3]{16}}$
$⇒{a}_{5}=\frac{10}{{5}^{\frac{2}{3}}}=\frac{10}{{\left({5}^{2}\right)}^{\frac{1}{3}}}=\frac{10}{\sqrt[3]{25}}$
Therefore, the first five terms of the sequence are $10,\frac{10}{\sqrt[3]{4}},\frac{10}{\sqrt[3]{9}},\frac{10}{\sqrt[3]{16}}$ and $\frac{10}{\sqrt[3]{25}}$

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