Pattern practice

Number Theory Level 2

What is the next number in each of the following sequences (they follow the same logic)?

1 , 4 , 27 , . . . 1 , 5 , 32 , . . . 1 , 4 , 108 , . . . 1, 4, 27, ... \\ 1, 5, 32, ... \\ 1, 4, 108, ...

256, 288, 27648 64, 96, 256 64, 96, 27648 256, 288, 256

Aditya Raj by Aditya Raj , 23, India

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1 solution

The first sequence is 1 1 , 2 2 , 3 3 , 4 4 , . . . 1^1,2^2,3^3,4^4,... . The second sequence is 1 1 , 1 1 + 2 2 , 1 1 + 2 2 + 3 3 , 1 1 + 2 2 + 3 3 + 4 4 , . . . 1^1,1^1+2^2,1^1+2^2+3^3,1^1+2^2+3^3+4^4,... . The third sequence is 1 1 , 1 1 × 2 2 , 1 1 × 2 2 × 3 3 , 1 1 × 2 2 × 3 3 × 4 4 , . . . 1^1,1^1\times 2^2,1^1\times 2^2\times 3^3,1^1\times 2^2\times 3^3\times 4^4,... . So the required numbers are 4 4 , 1 1 + 2 2 + 3 3 + 4 4 , 1 1 × 2 2 × 3 3 × 4 4 4^4,1^1+2^2+3^3+4^4,1^1\times 2^2\times 3^3\times 4^4 or 256 , 288 , 27648 \boxed {256,288,27648} .

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