Question

A radioactive substance decreases in the amount of grams by one-third each year. If the starting amount of the substance in a rock is 1.452 g, write a recursive formula for a sequence that models the amount of the substance left after then end of each year? A) aₙ = (1/3) * aₙ₋₁

B) aₙ = (2/3) * aₙ₋₁
C) aₙ = aₙ₋₁ - (1/3)
D) aₙ = aₙ₋₁ + (2/3)

Answer 1

The recursive formula to model a radioactive substance that decreases by one-third each year is B) an = (2/3) * an-1.

Step-by-step explanation:

A radioactive substance that decreases in amount by one-third each year is an example of exponential decay. The starting amount is given as 1.452 g, and after each year, only two-thirds of the substance remains because one-third is lost to radioactive decay.

To model the amount of the substance left after each year using a recursive formula, we can define the first term a1 as 1.452 g (the initial amount) and each subsequent term by multiplying the previous term by two-thirds, reflecting the exponential decay process.

Thus, the recursive formula to model this decay is B) an = (2/3) * an-1. This represents that each year the amount remaining is 2/3 of the previous year's amount.