Does the sequence contain itself?

Number Theory Level 2

The following is the start of the sequence of the digit sum of the positive integers: $1, 2, 3, 4, 5, 6, 7, 8, 9, \color{#D61F06}1\color{#333333}, 2, 3, 4, 5, 6, 7, 8, 9, 10, \color{#D61F06}2\color{#333333}, 3, 4, 5, 6, 7, 8, 9, 10, 11, \color{#D61F06}3\color{#333333}, 4, 5, 6, 7,\ldots.$ Most terms of the sequence are $1$ more than the previous term but some are not. The first three of those $({\color{#D61F06}1, 2, 3})$ are shown in $\color{#D61F06}\text{red}\color{#333333}$ above, which correspond to $10, 20, 30$ in the original sequence, respectively.

Create a new sequence of these $\color{#D61F06}\text{red}\color{#333333}$ numbers.

Is this new sequence the same as the original?

Yes No

1 solution

Jeremy Galvagni
Apr 25, 2018

The only time the sequence drops is when the ones digit goes from 9 to 0. This happens every 10th number. So the red numbers come from multiples of 10. Divide the multiples of 10 by 10 and you get the sequence of positive integers. But these are the numbers that defined the original sequence! So $\boxed{Yes}$ you get the original sequence back.

Does the sequence contain itself?

1 solution

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