For a- R- -a-1- let n-lim1-2- -nn7-31an-12-1an-22-1an-n2-54-Then the possible values of a is-are

N→∞lim(1^1/3+2^1/3+ ... +n^1/3n^7/3n^2(1(a+1n)^2+1(a+2n)^2+...+1(a+nn)^2))=54 ⇒n→∞lim(1n∑_r=1^n(rn)^1/31n∑_r=1^n(1a+rn)^2)=54 ⇒∫_0^1 x^1/3dx∫_0^1(1/a+x)^2dx=3 ...

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Standard XII

Mathematics

Right Hand Derivative

For a∈ R, |a...

Question

For $a \in R, | a | > 1,$ let
$lim n \to \infty ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{1 + \sqrt[3]{2} + . . . + \sqrt[3]{n}}{n^{\frac{7}{3}} (\frac{1}{(a n + 1)^{2}} + \frac{1}{(a n + 2)^{2}} + . . . + \frac{1}{(a n + n)^{2}})} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = 54.$

Then the possible value(s) of $a$ is/are

- 9

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- 6

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7

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Solution

The correct option is D 8
$lim n \to \infty ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{1^{\frac{1}{3}} + 2^{\frac{1}{3}} + . . . + n^{\frac{1}{3}}}{\frac{n^{\frac{7}{3}}}{n^{2}} ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{1}{{(a + \frac{1}{n})}^{2}} + \frac{1}{{(a + \frac{2}{n})}^{2}} + . . . + \frac{1}{{(a + \frac{n}{n})}^{2}} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = 54$

$\Rightarrow lim n \to \infty ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{\frac{1}{n} n \sum r = 1 {(\frac{r}{n})}^{\frac{1}{3}}}{\frac{1}{n} n \sum r = 1 {⎛ ⎜ ⎜ ⎝ \frac{1}{a + \frac{r}{n}} ⎞ ⎟ ⎟ ⎠}^{2}} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = 54$

$\Rightarrow \frac{1 \int 0 x^{\frac{1}{3}} d x}{1 \int 0 {(\frac{1}{a + x})}^{2} d x} = \frac{\frac{3}{4} [x^{\frac{4}{3}}]_{0}^{1}}{- {[\frac{1}{a + x}]}_{0}^{1}} = 54$

$\Rightarrow \frac{3}{4} a (a + 1) = 54 \Rightarrow a^{2} + a - 72 = 0$
$∴ a = - 9, 8$

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For a∈R, |a|>1, let limn→∞⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1+3√2+...+3√nn73(1(an+1)2+1(an+2)2+...+1(an+n)2)⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠=54. Then the possible value(s) of a is/are