For some \(n\) and \(k\) let \(a_{1}, a_{2}, ..., a_{n}\) be a sequence where:

\(a_{i}\) is a positive integer

\(a_{1} = k\)

\(a_{i+1} = 2 \times a_{i}\)

Digit sums of all \(a_{i }\) are equal (base \(10\))

Can we construct such sequence for any positive integer \(n\) (\(k\) is of your choice)?

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