Maths-
General
Easy

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

  1. 27 over 16
  2. 9 over 4
  3. 16 over 27
  4. 4 over 9

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 equalslimit 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

    Book A Free Demo

    +91

    Grade*

    Related Questions to study

    General
    Maths-

    If fraction numerator b over denominator c plus a end fraction plus fraction numerator c over denominator a plus b end fraction equals 1then

    If fraction numerator b over denominator c plus a end fraction plus fraction numerator c over denominator a plus b end fraction equals 1then

    Maths-General
    General
    maths-

    Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator 1 over denominator sin squared begin display style space end style x end fraction minus fraction numerator 1 over denominator sinh squared space x end fraction equals

    Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator 1 over denominator sin squared begin display style space end style x end fraction minus fraction numerator 1 over denominator sinh squared space x end fraction equals

    maths-General
    General
    Maths-

    The quadratic equation whose roots are I and m where l =lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 3 sin space theta minus 4 sin cubed space theta over denominator theta end fraction close parentheses m equals lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 2 tan space theta over denominator theta open parentheses 1 minus tan squared space theta close parentheses end fraction close parentheses is

    Given, the quadratic equation whose roots are I and m where
    l =lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 3 sin space theta minus 4 sin cubed space theta over denominator theta end fraction close parentheses m equals lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 2 tan space theta over denominator theta open parentheses 1 minus tan squared space theta close parentheses end fraction close parentheses
    l equals limit as theta rightwards arrow 0 of open parentheses fraction numerator 3 space sin theta minus 4 sin cubed theta over denominator theta end fraction close parentheses space space space space space space space space space space space space space space space space space space space space m equals limit as theta rightwards arrow 0 of open parentheses fraction numerator 2 tan theta over denominator theta left parenthesis 1 minus tan squared theta right parenthesis end fraction close parentheses
space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator sin space 3 theta over denominator theta end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator tan space 2 theta over denominator theta end fraction close parentheses
space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator sin space 3 theta over denominator 3 theta end fraction close parentheses cross times 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as theta rightwards arrow 0 of space open parentheses fraction numerator sin 2 theta over denominator theta cos 2 theta end fraction close parentheses
space equals 1 cross times 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator sin 2 theta over denominator 2 theta end fraction close parentheses cross times 2 cross times fraction numerator 1 over denominator cos 2 theta end fraction
space equals 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 2
l equals 3 space a n d space m equals 2
t h e space q u a d t r a t i c space e q u a t i o n space w h e n space t h e space r o o t s space a r e space g i v e n
equals x squared minus left parenthesis l plus m right parenthesis x plus l m
equals x squared minus left parenthesis 3 plus 2 right parenthesis x plus left parenthesis 3 cross times 2 right parenthesis
equals x squared minus 5 x plus 6 space

    The quadratic equation whose roots are I and m where l =lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 3 sin space theta minus 4 sin cubed space theta over denominator theta end fraction close parentheses m equals lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 2 tan space theta over denominator theta open parentheses 1 minus tan squared space theta close parentheses end fraction close parentheses is

    Maths-General
    Given, the quadratic equation whose roots are I and m where
    l =lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 3 sin space theta minus 4 sin cubed space theta over denominator theta end fraction close parentheses m equals lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 2 tan space theta over denominator theta open parentheses 1 minus tan squared space theta close parentheses end fraction close parentheses
    l equals limit as theta rightwards arrow 0 of open parentheses fraction numerator 3 space sin theta minus 4 sin cubed theta over denominator theta end fraction close parentheses space space space space space space space space space space space space space space space space space space space space m equals limit as theta rightwards arrow 0 of open parentheses fraction numerator 2 tan theta over denominator theta left parenthesis 1 minus tan squared theta right parenthesis end fraction close parentheses
space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator sin space 3 theta over denominator theta end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator tan space 2 theta over denominator theta end fraction close parentheses
space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator sin space 3 theta over denominator 3 theta end fraction close parentheses cross times 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as theta rightwards arrow 0 of space open parentheses fraction numerator sin 2 theta over denominator theta cos 2 theta end fraction close parentheses
space equals 1 cross times 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator sin 2 theta over denominator 2 theta end fraction close parentheses cross times 2 cross times fraction numerator 1 over denominator cos 2 theta end fraction
space equals 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 2
l equals 3 space a n d space m equals 2
t h e space q u a d t r a t i c space e q u a t i o n space w h e n space t h e space r o o t s space a r e space g i v e n
equals x squared minus left parenthesis l plus m right parenthesis x plus l m
equals x squared minus left parenthesis 3 plus 2 right parenthesis x plus left parenthesis 3 cross times 2 right parenthesis
equals x squared minus 5 x plus 6 space
    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

    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

    Maths-General
    General
    maths-

    Let f : [-10, 10] →R, where f(x) = sin x + open square brackets fraction numerator x to the power of 2 end exponent over denominator a end fraction close square brackets be an odd function. Then set of values of parameter a is/are

    Let f : [-10, 10] →R, where f(x) = sin x + open square brackets fraction numerator x to the power of 2 end exponent over denominator a end fraction close square brackets be an odd function. Then set of values of parameter a is/are

    maths-General
    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
    General
    maths-

    If the function not stretchy integral subscript 0 end subscript superscript x end superscript blankf(x) dt → 5 as |x| → I, then the value of ‘a’ so that the equation 2x + not stretchy integral subscript 0 end subscript superscript x end superscript blankf(t) dt = a has at least two roots of opposite signs in (-1, 1) is

    If the function not stretchy integral subscript 0 end subscript superscript x end superscript blankf(x) dt → 5 as |x| → I, then the value of ‘a’ so that the equation 2x + not stretchy integral subscript 0 end subscript superscript x end superscript blankf(t) dt = a has at least two roots of opposite signs in (-1, 1) is

    maths-General
    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

    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

    Maths-General
    General
    chemistry-

    On Bohr’s stationary orbits -

    On Bohr’s stationary orbits -

    chemistry-General
    General
    chemistry-

    For a valid Bohr orbit, its cicumfrence should be -

    For a valid Bohr orbit, its cicumfrence should be -

    chemistry-General
    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
    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

    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

    Maths-General
    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
    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

    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
    open parentheses fraction numerator c plus a plus b over denominator c end fraction close parentheses open parentheses fraction numerator b plus c minus a over denominator b end fraction close parentheses
equals fraction numerator b c plus c squared minus a c plus a b plus a c minus a squared plus b squared plus b c minus a b over denominator c b end fraction
equals fraction numerator 2 b c plus c squared plus b squared minus a squared over denominator c b end fraction
equals fraction numerator 2 b c plus 2 b c space cos space A over denominator b c end fraction space space left parenthesis b y space cos i n e space f o r m u l a comma space a squared equals b squared plus c squared minus 2 b c space cos A right parenthesis
equals 2 plus 2 space cos A
p u t t i n g space A equals 60 degree
equals 2 plus 2 space cos space 60 degree
equals 2 plus 2 cross times 1 half
equals 2 plus 1
equals 3

    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

    Maths-General
    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
    open parentheses fraction numerator c plus a plus b over denominator c end fraction close parentheses open parentheses fraction numerator b plus c minus a over denominator b end fraction close parentheses
equals fraction numerator b c plus c squared minus a c plus a b plus a c minus a squared plus b squared plus b c minus a b over denominator c b end fraction
equals fraction numerator 2 b c plus c squared plus b squared minus a squared over denominator c b end fraction
equals fraction numerator 2 b c plus 2 b c space cos space A over denominator b c end fraction space space left parenthesis b y space cos i n e space f o r m u l a comma space a squared equals b squared plus c squared minus 2 b c space cos A right parenthesis
equals 2 plus 2 space cos A
p u t t i n g space A equals 60 degree
equals 2 plus 2 space cos space 60 degree
equals 2 plus 2 cross times 1 half
equals 2 plus 1
equals 3
    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 (mu equals 1) will be equal to

    In the case of refraction if CD is the refracted wave front and v1 and v2 are the speed of light in the two media, then in the time the wavelets from B reaches C, the wavelet from A will reach D, such that

    t equals fraction numerator B C over denominator v subscript a end subscript end fraction equals fraction numerator A D over denominator v subscript g end subscript end fraction rightwards double arrow fraction numerator B C over denominator A D end fraction equals fraction numerator v subscript a end subscript over denominator v subscript g end subscript end fraction .....(i)
    But in capital delta A C B comma B C equals A C sin invisible function application theta .....(ii)
    while in capital delta A C D comma A D equals A C sin invisible function application phi to the power of ´ end exponent .....(iii)
    From equations (i), (ii) and (iii) fraction numerator v subscript a end subscript over denominator v subscript g end subscript end fraction equals fraction numerator sin invisible function application theta over denominator sin invisible function application phi to the power of ´ end exponent end fraction
    Also mu proportional to fraction numerator 1 over denominator v end fraction rightwards double arrow fraction numerator v subscript a end subscript over denominator v subscript g end subscript end fraction equals fraction numerator mu subscript g end subscript over denominator mu subscript a end subscript end fraction equals fraction numerator sin invisible function application theta over denominator sin invisible function application phi to the power of ´ end exponent end fraction rightwards double arrow mu subscript g end subscript equals fraction numerator sin invisible function application theta over denominator sin invisible function application phi to the power of ´ end exponent end fraction

    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 (mu equals 1) will be equal to

    physics-General
    In the case of refraction if CD is the refracted wave front and v1 and v2 are the speed of light in the two media, then in the time the wavelets from B reaches C, the wavelet from A will reach D, such that

    t equals fraction numerator B C over denominator v subscript a end subscript end fraction equals fraction numerator A D over denominator v subscript g end subscript end fraction rightwards double arrow fraction numerator B C over denominator A D end fraction equals fraction numerator v subscript a end subscript over denominator v subscript g end subscript end fraction .....(i)
    But in capital delta A C B comma B C equals A C sin invisible function application theta .....(ii)
    while in capital delta A C D comma A D equals A C sin invisible function application phi to the power of ´ end exponent .....(iii)
    From equations (i), (ii) and (iii) fraction numerator v subscript a end subscript over denominator v subscript g end subscript end fraction equals fraction numerator sin invisible function application theta over denominator sin invisible function application phi to the power of ´ end exponent end fraction
    Also mu proportional to fraction numerator 1 over denominator v end fraction rightwards double arrow fraction numerator v subscript a end subscript over denominator v subscript g end subscript end fraction equals fraction numerator mu subscript g end subscript over denominator mu subscript a end subscript end fraction equals fraction numerator sin invisible function application theta over denominator sin invisible function application phi to the power of ´ end exponent end fraction rightwards double arrow mu subscript g end subscript equals fraction numerator sin invisible function application theta over denominator sin invisible function application phi to the power of ´ end exponent end fraction