Concept:This is the fundamental universal mathematical equation for exponential radioactive decay based on discrete half-life intervals.
Solution:- After 1 half-life, you have \( \frac{1}{2} \) of the initial amount.
- After 2 half-lives, you have \( \frac{1}{2} \times \frac{1}{2} = \left(\frac{1}{2}\right)^2 \) of the initial amount.
- Generalizing this pattern, after \(n\) half-lives, the remaining amount is strictly multiplied by the fraction \( \left(\frac{1}{2}\right)^n \).
- Yielding the standard formula: $$ N = N_0 \left(\frac{1}{2}\right)^n $$
Why other options are incorrect:Option A describes exponential growth (like bacterial replication), not decay. Options C and D apply the exponent incorrectly to the initial quantity (\(N_o\)) itself, which makes zero mathematical sense.
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