For a\inR, | a | > 1, let \underset{n \rightarrow \infty}{Lim} \left(\right. \frac{1 + \sqrt[3]{2} + . ....... + \sqrt[3]{n}}{n^{7 / 3} \left(\right. \frac{1}{\left(\left(\right. an + 1 \left.\right)\right)^{2}} + \frac{1}{\left(\left(\right. an + 2 \left.\right)\right)^{2}} + . ...... + \frac{1}{\left(\left(\right. an + n \left.\right)\right)^{2}} \left.\right)} \left.\right) = 54. Then the possible value(s) of a is/are

Engineering

Mathematics

Introduction to Definite Integral

Question

For a $\in$ R, | a | > 1, let $\underset{n \to \infty}{Lim} (\frac{1 + \sqrt[3]{2} + . ....... + \sqrt[3]{n}}{n^{7 / 3} (\frac{1}{{(an + 1)}^{2}} + \frac{1}{{(an + 2)}^{2}} + . ...... + \frac{1}{{(an + n)}^{2}})})$ = 54. Then the possible value(s) of a is/are

–6

–9

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$\underset{n \to \infty}{Lim} \frac{n^{\frac{1}{3}} (\sum_{r = 1}^{n} {(\frac{r}{n})}^{1 / 3})}{n^{\frac{7}{3}} (\sum_{r = 1}^{n} \frac{1}{{(an + r)}^{2}})} = 54 \Rightarrow \underset{n \to \infty}{Lim} (\frac{\frac{1}{n} \sum_{r = 1}^{n} {(\frac{r}{n})}^{1 / 3}}{\frac{1}{n} \sum_{r = 1}^{n} \frac{1}{{(a + \frac{r}{n})}^{2}}}) = 54$

$\Rightarrow \frac{\int_{0}^{1} x^{\frac{1}{3}} dx}{\int_{0}^{1} \frac{dx}{{(a + x)}^{2}}} = 54 \Rightarrow \frac{\frac{3}{4}}{\frac{1}{a (a + 1)}} = 54 \Rightarrow a (a + 1) = 72$

⇒ a² + a – 72 = 0 ⇒ a = – 9, 8.