Based on the answer provided, it seems the writer wanted you to assume that the energy loss per plank is constant. This is not the same as the bullet losing 1/nth1/nth of its velocity per plank (however, the fact that the question does not mention this assumption arguably makes the question ambiguous).
With this assumption, the energy loss becomes
ΔE = 1/2 mv2 − 1/2 m (v−v/n) 2
and the number of planks NN becomes
N = 1/2mv2 /ΔE = n2/ 2n−1
Otherwise, if you assume that the bullet loses 1/nth1/nth of its velocity per plank, then the answer is N=∞
