Question

72–75. {Use of Tech} Practical sequences
Consider the following situations that generate a sequence

c.Find a recurrence relation that generates the sequence.

Radioactive decay
A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let Mₙ be the mass of the radioactive material at the end of the nᵗʰ decade, where the initial mass of the material is M₀ = 20g.

Accepted Answer

Identify the nature of the sequence: Since the material loses 50% of its mass every 10 years, the mass at the end of each decade is half of the mass at the previous decade. This indicates a geometric sequence where each term is multiplied by a constant ratio. Define the recurrence relation: The mass at the end of the (n+1)ᵗʰ decade, denoted as $$M_{n+1}$$, depends on the mass at the end of the nᵗʰ decade, $$M_n. $$Since the mass halves every decade, the relation can be written as $$M_{n+1} = r 	imes M_n$$, where $$r is $$the common ratio. Determine the common ratio $$r$$: Because the material transmutes 50% of its mass, the remaining mass is 50%, so $$r = 0.5. $$Write the full recurrence relation: Substitute the value of $$r$$ into the relation to get $$M_{n+1} = 0.5 	imes M_n. $$Include the initial condition: The problem states the initial mass is $$M_0 = 20$ grams, which is necessary to generate the sequence starting point.

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Key Concepts

Recurrence Relations

Exponential Decay

Initial Conditions in Sequences