1. Consider the sequence of natural numbers $1, 4, 7, \ldots, 2014, \ldots$.

(a) Investigate whether the numbers 826 and 2466 are terms of the sequence.

(b) Determine the three-digit numbers in the sequence that become six times larger when a digit is added in front.
Solution.

(a) The numbers in the sequence are of the form $3 k+1, k \in \mathbb{N}$. (1p)

Since $826=3 \cdot 275+1$, it is part of the sequence, but $2466=3 \cdot 821+3$ is not part of the sequence. (2pp)

(b) Let $\overline{a b c}$ be the 3-digit number we are looking for. We seek $\overline{d a b c}=6 \cdot \overline{a b c}$, which means $1000 \cdot d+\overline{a b c}=6 \cdot \overline{a b c}$. (1p)

It follows that $1000 \cdot d=5 \cdot \overline{a b c}, 200 \cdot d=\overline{a b c}, d \in\{1,2,3,4\}$ and the corresponding numbers are $200,400,600$, 800. (2p)

Only 400 is a term in the sequence. (1p)