Using multiplication factor \(k\) and considering generation wise growth of neutrons,

(i) show that \(N(t)=N(0) \exp ((k-1) t / \ell)\).

Here \(t\) is the time between first and \(n\)th generation and \(N(0)\) is the number of neutrons in first generation.

(ii) Taking \(\ell=10^{-04} \mathrm{~s}, N(0)=1\) and reactivity \(=1 \mathrm{mk}\), determine the number of neutrons generated in 100th, 500th, 1000th, and 10,000th generations of neutrons.