If a bullet loses 1/nth of its velocity while passing through a plank, then no of planks required to stop the bullet is?

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=∞