Cayley tree reachable nodes proof

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Imagine being in the central node of a Cayley Tree graph like this one: Cayle Tree, K =3 and D = 5

For a number of reachable t steps (t<D) prove that the reacheable nodes from the center equal:

$$ N_t = \frac {K(K-1)^t -2}{K-2}$$

P.S. Thanks!