Sum of the square of natural number is given as \(% MathType!MTEF!2!1!+-% feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8% qacaWGtbGaeyypa0ZaaSaaa8aabaWdbiaad6gadaqadaWdaeaapeGa% amOBaiabgUcaRiaaigdaaiaawIcacaGLPaaadaqadaWdaeaapeGaaG% Omaiaad6gacqGHRaWkcaaIXaaacaGLOaGaayzkaaaapaqaa8qacaaI% 2aaaaaaa!4322!S = \frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}\)Sum of the cube of natural number is given as,C = \(% MathType!MTEF!2!1!+-% feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8% qadaqadaWdaeaapeWaaSaaa8aabaWdbiaad6gadaqadaWdaeaapeGa% amOBaiabgUcaRiaaigdaaiaawIcacaGLPaaaa8aabaWdbiaaikdaaa% aacaGLOaGaayzkaaWdamaaCaaaleqabaWdbiaaikdaaaaaaa!3EFC!{\left( {\frac{{n\left( {n + 1} \right)}}{2}} \right)^2}\)Hence,\(% MathType!MTEF!2!1!+-% feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8% qadaWcaaWdaeaapeGaam4qaaWdaeaapeGaam4uaaaacqGH9aqpdaWc% aaWdaeaapeWaaSaaa8aabaWdbiaad6gadaqadaWdaeaapeGaamOBai% abgUcaRiaaigdaaiaawIcacaGLPaaadaqadaWdaeaapeGaaGOmaiaa% d6gacqGHRaWkcaaIXaaacaGLOaGaayzkaaaapaqaa8qacaaI2aaaaa% WdaeaapeWaaeWaa8aabaWdbmaalaaapaqaa8qacaWGUbWaaeWaa8aa% baWdbiaad6gacqGHRaWkcaaIXaaacaGLOaGaayzkaaaapaqaa8qaca% aIYaaaaaGaayjkaiaawMcaa8aadaahaaWcbeqaa8qacaaIYaaaaaaa% kiabg2da9maalaaapaqaa8qacaaIZaaapaqaa8qacaaIYaaaaiabgE% na0oaalaaapaqaa8qacaWGUbGaaiiOamaabmaapaqaa8qacaWGUbGa% ey4kaSIaaGymaaGaayjkaiaawMcaaaWdaeaapeWaaeWaa8aabaWdbi% aaikdacaWGUbGaey4kaSIaaGymaaGaayjkaiaawMcaaaaaaaa!5DEA!\frac{C}{S} = \frac{{\frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}}}{{{{\left( {\frac{{n\left( {n + 1} \right)}}{2}} \right)}^2}}} = \frac{3}{2} \times \frac{{n\;\left( {n + 1} \right)}}{{\left( {2n + 1} \right)}}\)? At n = 1, \(% MathType!MTEF!2!1!+-% feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8% qadaWcaaWdaeaapeGaam4qaaWdaeaapeGaam4uaaaacqGH9aqpcaaI% WaGaeyypa0ZaaSaaa8aabaWdbiaaicdaa8aabaWdbiaaigdaaaaaaa!3C8E!\frac{C}{S} = 0 = \frac{0}{1}\)? At n = 2, \(% MathType!MTEF!2!1!+-% feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8% qadaWcaaWdaeaapeGaam4qaaWdaeaapeGaam4uaaaacqGH9aqpcaaI% Xaaaaa!39C6!\frac{C}{S} = 1\)Similarly at n = 8, \(% MathType!MTEF!2!1!+-% feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8% qadaWcaaWdaeaapeGaam4qaaWdaeaapeGaam4uaaaacqGH9aqpcaGG% GcWaaSaaa8aabaWdbiaaiodaa8aabaWdbiaaikdaaaGaey41aq7aaS% aaa8aabaWdbiaaiIdacaGGGcWaaeWaa8aabaWdbiaaiIdacqGHRaWk% caaIXaaacaGLOaGaayzkaaaapaqaa8qadaqadaWdaeaapeGaaGOmai% abgEna0kaaiIdacqGHRaWkcaaIXaaacaGLOaGaayzkaaaaaiabg2da% 9maalaaapaqaa8qacaaIXaGaaGimaiaaiIdaa8aabaWdbiaaigdaca% aI3aaaaaaa!5029!\frac{C}{S} = \;\frac{3}{2} \times \frac{{8\;\left( {8 + 1} \right)}}{{\left( {2 \times 8 + 1} \right)}} = \frac{{108}}{{17}}\)