Maths-
General
Easy
Question
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](data:image/png;base64,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)
![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](data:image/png;base64,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)
![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](data:image/png;base64,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)
=![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 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)
Related Questions to study
Maths-
If
then
If
then
Maths-General
maths-
maths-General
Maths-
The quadratic equation whose roots are I and m where l =
is
The quadratic equation whose roots are I and m where l =
is
Maths-General
Maths-
In ![straight triangle A B C comma left parenthesis a plus b plus c right parenthesis open parentheses tan invisible function application A over 2 plus tan invisible function application B over 2 close parentheses equals](data:image/png;base64,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)
In ![straight triangle A B C comma left parenthesis a plus b plus c right parenthesis open parentheses tan invisible function application A over 2 plus tan invisible function application B over 2 close parentheses equals](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQgAAAAmCAYAAAA4JwUNAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAY00gKyAAABwJJREFUeNrtXXFkXVcYP66IiCgVU1MVIqpqotRURUyIqaiKERMxUyNmaqpCVcRElJqZiilVUVVVamKqalRU1EyZeaKqwtT+iIowFTER4e78lu/JzXnnvXvOfee8d+67349P3ru5753zne/c737fd37nPiEYjEp8I+UnHoZcYJ7sxWA0BINSSlLaAuzb5QLZYVvKrpS/pfwl5bGUDzXnwU4rUj7hqcvwjU4pb6WcCrBvn0vZCdRxuUYfOekk5qTcrXL+R2S3Lp7CDJ/AJLwVYL8O013ymZTRAtgBOj5Qjn2mOaamGnM8hRm+8IGU91KOBti321LGpXwpilEb+U7KVeXYPSnnanwG6ccm2ZHBcI5pKfcD7NdZKU/o9RHKx1sdjyiKiCjdWxBm9RecN8NTmeEDr8VegTIkoN7wh5RjiWN/SjnR4rZYlRKTbKVEDkkMSHnDU5nhGselbAQaak8px66L1l7Wi8gpAO1ShqX8TjYywXoBHCijwbgoahfAmoETFC2owHLe4xa2xRkpS8qxCSnfG37+AdmTwXAGFMC+DqxPLxJhtiqtvNw5TrWEJL6Qcsfw81+RPRkWQEh6g4fh/zHQhecoAo4E1M9JKTdr/P9hYP11iXlNBLBAjsMEqFc85alujn7K4Rh7+I3GJAnUH0JZHsNyHTgPXSl3yZstap9FsV+UhE3GxB5pqt3w812igfUkFEhA+Ryq4zvSlmdMKaW9lIdhsP6l80EMOWxwQTSbGYj+RoFMwNMUvofaPyzxnU85B1yN1RZ1EP8kUqltipZsuCkR2dM7IvLkw/Q3ywRKo8eaUkqnKAweTPTjGDmfuzXaP0UOopnoo/ELLb8/nXgfp0zYIiLPeu80opErUmbp9ayoXG5Kgwk91oRSCkbZnYw6IOK45GFsYotzR+ku0Kj2TPCtOFiTiRvYdrMRN2nMQ9QxM7rFHnGmk9530vtui+8wocemUUqPU9qRtWL9q9CTfyJyeuvkbRHF9HgyAHScpvbeUdj4vEoa5crgXeQc10i/RSUVAztxyeD71dUDFRekvKQ2EPIPVOn3GNkRui+Lg8SnEC+evOvt3UHg4p5Qjo3TcROY0mPTKKU/iPq2+G5WKe6gb/N00USJfvgwwCPS/2qivR8to4rY0jmAPwBC0SHSf0rJ69tpbOqNIBDpnaTXqLy/0nx2jJxVT+K85ZxHEKHrberwYwNnaJW36yrgKmzosWmU0teUw2fFrubYOF20atQy4umCXU1MpjIOKReoy/ZukPOzyVNdpBiRJvfFZ4cMzstzihGi3l4jiFpOwKToZ0qPTaOURhSaCccO4jlFOEksp6QYWT1vtYpyR4qDqKe9DZG+suPDQejOjS2+M65TXI9l6Ho3xUGYpBHl2oIONvTYNEppu+Vd1jTFUJf0ko7KtYc+Qw5Jl4IteWjvY1G5hKmDqxRjhiKk3SoTN270BG5QBBG63l5SDNNCZDflXB2a/9nQY00opVtiv1CaBc80BSSVRDJQxam5mGTQUbcMe0XYbcs1bQ9p0qJhjWipTgdxm/LsDkd30rw4iDzo7aWdOZq4JkAKMascs6XHmlBKF4T98moSumXONcW4uFB/9mQAnY64e2P518cqBrgNJtt9L4mDm3/ilFQkqhKduQy1Q3MQedbbeTtgKZaEORkqovN76X0WeqwJpRR1ARBWriXyalzc58l5DCsOCAMzmlIzmaO7QETfj4r0U08GWKRQtNzPo3TsokeDl6jmg2gNxdDLmijKhihVqpJSviGblvXCuL4TB1l/eXYQedbbeTu/ZCyUlMPZLPRYU0opVjHukefGue/pjj+iiVBUBwGg8KkWXaepzzPkcNaFn2cd4nsvJPLVkvD/TEWsHi1TeyuJyZyMMl5YTKgh0kM9p59SM9xpX1K9pcwtCdlBmCLPercauc0ZdLv8TDZrFek5frpVqt0A+zlAN4JNsU9mm2hx27jSeYddQSWQTrwSetYlnnXA27336g6TmuMhbdYqY5lC/HLaepKc23gL28eFzm2iQZu18gakJ0d4GDJhTZhxJ5qNHhHexrfQdMYq4wZPaYZLgHp+Lid93S6gfWx0RorND4xhOAW4Gnl4juFZ0fzt+6HrPClqPwKBwbAGCGr3A+8jVpuwajBQILtk0fmh4IfWMhwD+1/WA+4f6iNYiv+0QDbJqjPqD/zYe4ZzrIjKTWwhoJculL4C2SKrzuBu8A/nMLwAlPaFwPqEOyH25nQWyA716Iw08RpPZYYPYHkspB/vxZI12K5tBbJBPTqDQbsl7J76xmBYAc/yCOXXs58UMJeuR2dEHbM8hRk+gbAWj8frD6AvdT0WLafIqjNYxG9F7c2SDIYTYEmtVLDQPs8oPz5gkIeC0SiAbHOLhyEXQEro7XdV/wOVU2QMEqw9oQAAAZZ0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bWkgbWF0aHZhcmlhbnQ9Im5vcm1hbCI+JiN4MjVCMzs8L21pPjxtaT5BPC9taT48bWk+QjwvbWk+PG1pPkM8L21pPjxtbz4sPC9tbz48bW8+KDwvbW8+PG1pPmE8L21pPjxtbz4rPC9tbz48bWk+YjwvbWk+PG1vPis8L21vPjxtaT5jPC9taT48bW8+KTwvbW8+PG1mZW5jZWQgc2VwYXJhdG9ycz0ifCI+PG1yb3c+PG1pPnRhbjwvbWk+PG1vPiYjeDIwNjE7PC9tbz48bWZyYWM+PG1pPkE8L21pPjxtbj4yPC9tbj48L21mcmFjPjxtbz4rPC9tbz48bWk+dGFuPC9taT48bW8+JiN4MjA2MTs8L21vPjxtZnJhYz48bWk+QjwvbWk+PG1uPjI8L21uPjwvbWZyYWM+PC9tcm93PjwvbWZlbmNlZD48bW8+PTwvbW8+PC9tYXRoPiOmrEMAAAAASUVORK5CYII=)
Maths-General
Maths-
Let f : [-10, 10] →R, where f(x) = sin x +
be an odd function. Then set of values of parameter a is/are
Let f : [-10, 10] →R, where f(x) = sin x +
be an odd function. Then set of values of parameter a is/are
Maths-General
physics-
Light wave enters from medium 1 to medium 2. Its velocity in 2nd medium is double from 1st. For total internal reflection the angle of incidence must be greater than
Light wave enters from medium 1 to medium 2. Its velocity in 2nd medium is double from 1st. For total internal reflection the angle of incidence must be greater than
physics-General
maths-
If the function
f(x) dt → 5 as |x| → I, then the value of ‘a’ so that the equation 2x +
f(t) dt = a has at least two roots of opposite signs in (-1, 1) is
If the function
f(x) dt → 5 as |x| → I, then the value of ‘a’ so that the equation 2x +
f(t) dt = a has at least two roots of opposite signs in (-1, 1) is
maths-General
Maths-
In ![straight triangle A B C comma fraction numerator a left parenthesis cos invisible function application B plus cos invisible function application C right parenthesis over denominator 2 left parenthesis b plus c right parenthesis end fraction equals](data:image/png;base64,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)
In ![straight triangle A B C comma fraction numerator a left parenthesis cos invisible function application B plus cos invisible function application C right parenthesis over denominator 2 left parenthesis b plus c right parenthesis end fraction equals](data:image/png;base64,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)
Maths-General
chemistry-
On Bohr’s stationary orbits -
On Bohr’s stationary orbits -
chemistry-General
chemistry-
For a valid Bohr orbit, its cicumfrence should be -
For a valid Bohr orbit, its cicumfrence should be -
chemistry-General
chemistry-
Arrange the following particles in increasing order of values of e/m ratio: Electron (e), proton (p), neutron (n) and α-particle (α) -
Arrange the following particles in increasing order of values of e/m ratio: Electron (e), proton (p), neutron (n) and α-particle (α) -
chemistry-General
Maths-
In ![straight triangle A B C comma open parentheses fraction numerator b minus c over denominator a end fraction close parentheses sin invisible function application open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses equals](data:image/png;base64,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)
In ![straight triangle A B C comma open parentheses fraction numerator b minus c over denominator a end fraction close parentheses sin invisible function application open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses equals](data:image/png;base64,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)
Maths-General
physics-
An antenna behaves as resonant circuit only when its length is
An antenna behaves as resonant circuit only when its length is
physics-General
Maths-
If ![A equals 60 to the power of ring operator text then the value of end text open parentheses 1 plus a over c plus b over c close parentheses open parentheses 1 plus c over b minus a over b close parentheses equals](data:image/png;base64,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)
If ![A equals 60 to the power of ring operator text then the value of end text open parentheses 1 plus a over c plus b over c close parentheses open parentheses 1 plus c over b minus a over b close parentheses equals](data:image/png;base64,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)
Maths-General
physics-
In the adjoining diagram, a wavefront AB, moving in air is incident on a plane glass surface XY. Its position CD after refraction through a glass slab is shown also along with the normals drawn at A and D. The refractive index of glass with respect to air (
) will be equal to
![](data:image/png;base64,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)
In the adjoining diagram, a wavefront AB, moving in air is incident on a plane glass surface XY. Its position CD after refraction through a glass slab is shown also along with the normals drawn at A and D. The refractive index of glass with respect to air (
) will be equal to
![](data:image/png;base64,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)
physics-General