Geometric sequences are really important when figuring out how radiation decays over time. They help us understand something called exponential decay. Let's break it down step by step.

1. Radioactive Decay:

   • When a radioactive substance breaks down, we can use a formula to see how much is left: $N(t) = N_0 \times (0.5)^{t/T}$
   • In this formula,
     • $N(t)$ is how much is left after a certain time $t$.
     • $N_0$ is the amount we started with.
     • $T$ is the half-life, which is the time it takes for half of the substance to decay.

2. Half-Life Example:

   • Let's look at Carbon-14, which has a half-life of about 5,730 years.
   • If we begin with 1,000 grams, after 5,730 years, we would have: $N(5730) = 1000 \times (0.5)^{1} = 500 \text{ grams}$
   • After another 5,730 years, which is a total of 11,460 years, we would have: $N(11460) = 1000 \times (0.5)^{2} = 250 \text{ grams}$

3. Real-world Importance:

   • Understanding these sequences is really useful!
   • They help scientists date ancient objects and manage nuclear waste safely.

In summary, geometric sequences play a big role in studying the breakdown of radioactive materials. They help us know how long it takes for things to decay, and why that's important for science and safety.