Question

1. professor curie is using the radioactive isotope francium-223 in her research. this isotope decays by half every 22-minute interval. if professor curie has 15 grams of francium-223, which explicit formula describes the amount of francium-223 remaining?

\\(a_n = 15(0.5)^{n-1}\\)

\\(a_n = 15(22)^{n-1}\\)

\\(a_n = 22(0.5)^{15-1}\\)

\\(a_n = 22(0.5)^{n-1}\\)

Explanation:

⚡ Using what you learned: Geometric Sequences

Step 1: Identify the sequence parameters

The problem describes radioactive decay, which is modeled by a geometric sequence where the amount of substance is halved at each interval.

• Initial amount (first term, \( a_1 \)): \( 15 \) grams
• Common ratio (\( r \)): \( 0.5 \) (since it decays by half)

Step 2: Apply the explicit formula

The explicit formula for a geometric sequence is:

$$ a_n = a_1(r)^{n-1} $$

Substituting the identified values:

$$ a_n = 15(0.5)^{n-1} $$

Answer:

\( a_n = 15(0.5)^{n-1} \)