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Pre-RMO 2014/2

The first term of a sequence is \(2014\). Each succeeding term is the sum of the cubes of the digits of the previous term. What is the \(2014^{\text{th}}\) term of the sequence?


This note is part of the set Pre-RMO 2014

Note by Pranshu Gaba
2 years, 11 months ago

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Let's define this sequence as \(a_1, a_2, a_3, ..., a_{2014}, ...\) where \(a_1 = 2014\). The sum of the cubes of the digits of 2014 is 73. \(a_2 = 73\). The sum of the cubes of the digits of 73 is 370. \(a_3 = 370\). The sum of the cubes of the digits of 370 is 370 again. From this, we yield \(a_k = a_{k+1}\) for \(k \geq 3\). Therefore, \(a_{2014} = a_3 = 370\). Therefore, the answer is 370. Sharky Kesa · 2 years, 11 months ago

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@Sharky Kesa That is an Armstrong number. :-) Sachin Vishwakarma · 1 year, 11 months ago

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@Sharky Kesa nice Vedant Saraswat · 2 years, 11 months ago

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@Sharky Kesa An elegant solution. Karthik Venkata · 2 years, 8 months ago

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\(1^{st}\) term \(=2014\)

\(2^{nd}\) term \(=73\)

\(3^{rd}\) term \(=370\)

\(4^{th}\) term \(=370\)

. . .

Same goes on and \(2014^{th}\) term \(=\boxed{370}\) Akshat Sharda · 2 years, 1 month ago

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First term is 2014,as per the given question the second term should be 2^3+1^3+4^3=73.Similarly third term will be 7^3 + 3^3 = 370.Now the rest of the terms as we go further comes out to be 370. So the ans is 370. Vivek Rao · 2 years, 11 months ago

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the answer is 370 because all the terms after the third term are 370 Abhishek Alva · 1 year, 2 months ago

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370 Gaurav Singh · 2 years, 1 month ago

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370 :). Luana De Moraes · 2 years, 9 months ago

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370 Karthi Kn · 2 years, 10 months ago

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370 Archit Agarwal · 2 years, 11 months ago

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370 Vedant Saraswat · 2 years, 11 months ago

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370 Zahra Y · 2 years, 11 months ago

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370 Shudipta _Cuet12 · 2 years, 11 months ago

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370 Subhajit Ghosh · 2 years, 11 months ago

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Here 2014=2^3+1^3+4^3=73 Second term =7^3+3^3=370 Third term=3^3+7^3=370 Therefore k>=3. Then a=370 Harish Krishnan · 2 years, 11 months ago

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370 Harish Krishnan · 2 years, 11 months ago

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Here I term is 2014. Second term is (2^3+0^3+1^3+4^3)=73. Third term is (7^4+3^3)=370. Now it is clear that ii and iii digits have only two natural no. And now if we sum the cube of digits then it will remain 370. Hence after iii term all the terms of this series will be 370. Hence answer is 370. Rahul Verma · 2 years, 11 months ago

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370 Pulkit Kapoor · 2 years, 11 months ago

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