Correct Answer - Option 1 : 108 / 17
Sum of the square of natural number is given as
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% feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
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% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8
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% amOBaiabgUcaRiaaigdaaiaawIcacaGLPaaadaqadaWdaeaapeGaaG
% Omaiaad6gacqGHRaWkcaaIXaaacaGLOaGaayzkaaaapaqaa8qacaaI
% 2aaaaaaa!4322!
S = \frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}\)
Sum of the cube of natural number is given as,
C = \(% MathType!MTEF!2!1!+-
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% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
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% qadaqadaWdaeaapeWaaSaaa8aabaWdbiaad6gadaqadaWdaeaapeGa
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{\left( {\frac{{n\left( {n + 1} \right)}}{2}} \right)^2}\)
Hence,
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% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
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% qadaWcaaWdaeaapeGaam4qaaWdaeaapeGaam4uaaaacqGH9aqpdaWc
% aaWdaeaapeWaaSaaa8aabaWdbiaad6gadaqadaWdaeaapeGaamOBai
% abgUcaRiaaigdaaiaawIcacaGLPaaadaqadaWdaeaapeGaaGOmaiaa
% d6gacqGHRaWkcaaIXaaacaGLOaGaayzkaaaapaqaa8qacaaI2aaaaa
% WdaeaapeWaaeWaa8aabaWdbmaalaaapaqaa8qacaWGUbWaaeWaa8aa
% baWdbiaad6gacqGHRaWkcaaIXaaacaGLOaGaayzkaaaapaqaa8qaca
% aIYaaaaaGaayjkaiaawMcaa8aadaahaaWcbeqaa8qacaaIYaaaaaaa
% kiabg2da9maalaaapaqaa8qacaaIZaaapaqaa8qacaaIYaaaaiabgE
% na0oaalaaapaqaa8qacaWGUbGaaiiOamaabmaapaqaa8qacaWGUbGa
% ey4kaSIaaGymaaGaayjkaiaawMcaaaWdaeaapeWaaeWaa8aabaWdbi
% aaikdacaWGUbGaey4kaSIaaGymaaGaayjkaiaawMcaaaaaaaa!5DEA!
\frac{C}{S} = \frac{{\frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}}}{{{{\left( {\frac{{n\left( {n + 1} \right)}}{2}} \right)}^2}}} = \frac{3}{2} \times \frac{{n\;\left( {n + 1} \right)}}{{\left( {2n + 1} \right)}}\)
∴ At n = 1, \(% MathType!MTEF!2!1!+-
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% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8
% qadaWcaaWdaeaapeGaam4qaaWdaeaapeGaam4uaaaacqGH9aqpcaaI
% WaGaeyypa0ZaaSaaa8aabaWdbiaaicdaa8aabaWdbiaaigdaaaaaaa!3C8E!
\frac{C}{S} = 0 = \frac{0}{1}\)
∴ At n = 2, \(% MathType!MTEF!2!1!+-
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% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
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% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
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% qadaWcaaWdaeaapeGaam4qaaWdaeaapeGaam4uaaaacqGH9aqpcaaI
% Xaaaaa!39C6!
\frac{C}{S} = 1\)
Similarly at n = 8, \(% MathType!MTEF!2!1!+-
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% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8
% qadaWcaaWdaeaapeGaam4qaaWdaeaapeGaam4uaaaacqGH9aqpcaGG
% GcWaaSaaa8aabaWdbiaaiodaa8aabaWdbiaaikdaaaGaey41aq7aaS
% aaa8aabaWdbiaaiIdacaGGGcWaaeWaa8aabaWdbiaaiIdacqGHRaWk
% caaIXaaacaGLOaGaayzkaaaapaqaa8qadaqadaWdaeaapeGaaGOmai
% abgEna0kaaiIdacqGHRaWkcaaIXaaacaGLOaGaayzkaaaaaiabg2da
% 9maalaaapaqaa8qacaaIXaGaaGimaiaaiIdaa8aabaWdbiaaigdaca
% aI3aaaaaaa!5029!
\frac{C}{S} = \;\frac{3}{2} \times \frac{{8\;\left( {8 + 1} \right)}}{{\left( {2 \times 8 + 1} \right)}} = \frac{{108}}{{17}}\)