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Let the sum of the squares of successive integers 0, 1, 2, ..., n - 1, n be denoted by S. Let the sum of the cubes of the same integers be denoted by C. It is desirable that C / S, as n increases in steps of 'unity' from 'zero', is given by the series:

\(\frac{0}{1},\frac{3}{3},\frac{9}{5},\frac{{18}}{7},\frac{{30}}{9},\) ...(for n = 0, 1, 2, 3, 4, ...).

What will this ratio be for n = 8?


1. 108 / 17
2. 103 / 17
3. 103 / 15
4. 100 / 15

1 Answer

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Best answer
Correct Answer - Option 1 : 108 / 17

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, \(% MathType!MTEF!2!1!+- % feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qadaWcaaWdaeaapeGaam4qaaWdaeaapeGaam4uaaaacqGH9aqpcaGG % GcWaaSaaa8aabaWdbiaaiodaa8aabaWdbiaaikdaaaGaey41aq7aaS % aaa8aabaWdbiaaiIdacaGGGcWaaeWaa8aabaWdbiaaiIdacqGHRaWk % caaIXaaacaGLOaGaayzkaaaapaqaa8qadaqadaWdaeaapeGaaGOmai % abgEna0kaaiIdacqGHRaWkcaaIXaaacaGLOaGaayzkaaaaaiabg2da % 9maalaaapaqaa8qacaaIXaGaaGimaiaaiIdaa8aabaWdbiaaigdaca % aI3aaaaaaa!5029! \frac{C}{S} = \;\frac{3}{2} \times \frac{{8\;\left( {8 + 1} \right)}}{{\left( {2 \times 8 + 1} \right)}} = \frac{{108}}{{17}}\)

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