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Maths-

General

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Question

$L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n open parentheses 1 cubed plus 2 cubed plus midline horizontal ellipsis times plus n cubed close parentheses squared over denominator open parentheses 1 squared plus 2 squared plus midline horizontal ellipsis plus n squared close parentheses cubed end fraction equals$

$\frac{27}{16}$
$\frac{9}{4}$
$\frac{16}{27}$
$\frac{4}{9}$

hint Hint:

To solve the given question first we will write the formula for the sum of cube of first natural numbers and sum of square of first natural numbers, then after solving we will use the L hospital theorem twice .

The correct answer is: $27 over 16$

$L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n open parentheses 1 cubed plus 2 cubed plus midline horizontal ellipsis times plus n cubed close parentheses squared over denominator open parentheses 1 squared plus 2 squared plus midline horizontal ellipsis plus n squared close parentheses cubed end fraction equals$ $limit as x rightwards arrow infinity of fraction numerator open square brackets n open parentheses open parentheses begin display style fraction numerator n left parenthesis n plus 1 right parenthesis over denominator 2 end fraction end style close parentheses squared close parentheses squared close square brackets over denominator open parentheses begin display style fraction numerator n open parentheses n plus 1 close parentheses open parentheses 2 n plus 1 close parentheses over denominator 6 end fraction end style close parentheses cubed end fraction$
= $equals limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator n to the power of 5 left parenthesis n plus 1 right parenthesis to the power of 4 over denominator 16 end fraction end style over denominator begin display style fraction numerator n cubed open parentheses n plus 1 close parentheses cubed open parentheses 2 n plus 1 close parentheses cubed over denominator 216 end fraction end style end fraction equals limit as x rightwards arrow infinity of fraction numerator n to the power of 5 left parenthesis n plus 1 right parenthesis to the power of 4 over denominator 16 end fraction cross times fraction numerator 216 over denominator n cubed open parentheses n plus 1 close parentheses cubed open parentheses 2 n plus 1 close parentheses cubed end fraction equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator n squared left parenthesis n plus 1 right parenthesis over denominator left parenthesis 2 n plus 1 right parenthesis cubed end fraction u sin g space L apostrophe s space H o s p i t a l space r u l e comma equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator d n squared over denominator d n end fraction end style left parenthesis n plus 1 right parenthesis plus n squared begin display style fraction numerator d left parenthesis n plus 1 right parenthesis over denominator d n end fraction end style over denominator begin display style fraction numerator d open parentheses 2 n plus 1 close parentheses cubed over denominator d n end fraction end style end fraction equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator 2 n left parenthesis n plus 1 right parenthesis plus n squared left parenthesis 1 plus 0 right parenthesis over denominator 3. left parenthesis 2 n plus 1 right parenthesis squared. left parenthesis 2 plus 0 right parenthesis end fraction equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator 2 n squared plus 2 n plus n squared over denominator 6 left parenthesis 2 n plus 1 right parenthesis squared end fraction equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator 3 n squared plus 2 n over denominator 6 left parenthesis 2 n plus 1 right parenthesis squared end fraction a g a i n space u sin g space L apostrophe s space H o s p i t a l space r u l e comma equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator d 3 n squared over denominator d n end fraction end style plus begin display style fraction numerator d 2 n over denominator d n end fraction end style over denominator 6 begin display style fraction numerator d left parenthesis 2 n plus 1 right parenthesis squared over denominator d n end fraction end style end fraction equals 27 over 2 space limit as x rightwards arrow infinity of fraction numerator 6 n plus 2 over denominator 24 left parenthesis 2 n plus 1 right parenthesis.2 end fraction equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator 6 plus begin display style 2 over x end style over denominator 48 plus begin display style 1 over x end style end fraction equals 27 over 2 cross times 6 over 48 equals 27 over 16$

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