Pre-RMO 2014/2

The first term of a sequence is 20142014. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the 2014th2014^{\text{th}} term of the sequence?


This note is part of the set Pre-RMO 2014

Note by Pranshu Gaba
5 years ago

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Let's define this sequence as a1,a2,a3,...,a2014,...a_1, a_2, a_3, ..., a_{2014}, ... where a1=2014a_1 = 2014. The sum of the cubes of the digits of 2014 is 73. a2=73a_2 = 73. The sum of the cubes of the digits of 73 is 370. a3=370a_3 = 370. The sum of the cubes of the digits of 370 is 370 again. From this, we yield ak=ak+1a_k = a_{k+1} for k3k \geq 3. Therefore, a2014=a3=370a_{2014} = a_3 = 370. Therefore, the answer is 370.

Sharky Kesa - 5 years ago

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That is an Armstrong number. :-)

Sachin Vishwakarma - 4 years ago

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nice

Vedant Saraswat - 5 years ago

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An elegant solution.

Karthik Venkata - 4 years, 9 months ago

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1st1^{st} term =2014=2014

2nd2^{nd} term =73=73

3rd3^{rd} term =370=370

4th4^{th} term =370=370

. . .

Same goes on and 2014th2014^{th} term =370=\boxed{370}

Akshat Sharda - 4 years, 2 months 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 - 5 years ago

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370

Pulkit Kapoor - 5 years 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 - 5 years ago

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370

Harish Krishnan - 5 years 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 - 5 years ago

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370

Subhajit Ghosh - 5 years ago

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370

Shudipta _cuet12 - 5 years ago

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370

Zahra Y - 5 years ago

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370

Vedant Saraswat - 5 years ago

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370

Archit Agarwal - 4 years, 11 months ago

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370

Karthi KN - 4 years, 11 months ago

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370 :).

Luana de Moraes - 4 years, 10 months ago

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370

gaurav singh - 4 years, 2 months ago

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the answer is 370 because all the terms after the third term are 370

abhishek alva - 3 years, 2 months ago

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370

Madhu Agarwal - 1 year, 2 months ago

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