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

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

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TopNewestLet'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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– Sachin Vishwakarma · 1 year, 11 months ago

That is an Armstrong number. :-)Log in to reply

– Vedant Saraswat · 2 years, 11 months ago

niceLog in to reply

– Karthik Venkata · 2 years, 8 months ago

An elegant solution.Log in to reply

\(1^{st}\) 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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