Back27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.
b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).
{1, 2, 4, 8, 16, ......}
27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.
c. Find an explicit formula for the nth term of the sequence.
{1, 2, 4, 8, 16, ......}
27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.
b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).
{1, 3, 9, 27, 81, ......}
27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.
c. Find an explicit formula for the nth term of the sequence.
{1, 3, 9, 27, 81, ......}
27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.
b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).
{-5, 5, -5, 5, ......}
27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.
c. Find an explicit formula for the nth term of the sequence.
{-5, 5, -5, 5, ......}
57–60. Heights of bouncing balls A ball is thrown upward to a height of hₒ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hₙ be the height after the nth bounce. Consider the following values of hₒ and r.
b. Find an explicit formula for the nth term of the sequence {hₙ}.
h₀ = 20,r = 0.5
57–60. Heights of bouncing balls A ball is thrown upward to a height of hₒ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hₙ be the height after the nth bounce. Consider the following values of hₒ and r.
b. Find an explicit formula for the nth term of the sequence {hₙ}.
h₀ = 30,r = 0.25
72–75. {Use of Tech} Practical sequences
Consider the following situations that generate a sequence
b.Find an explicit formula for the terms of 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.
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.
72–75. {Use of Tech} Practical sequences
Consider the following situations that generate a sequence
d.Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist.
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.
72–75. {Use of Tech} Practical sequences
Consider the following situations that generate a sequence
b.Find an explicit formula for the terms of the sequence.
Drug elimination
Jack took a 200-mg dose of a pain killer at midnight. Every hour, 5% of the drug is washed out of his bloodstream. Let dₙ be the amount of drug in Jack’s blood n hours after the drug was taken, where d₀ = 200mg.
72–75. {Use of Tech} Practical sequences
Consider the following situations that generate a sequence
c.Find a recurrence relation that generates the sequence.
Drug elimination
Jack took a 200-mg dose of a pain killer at midnight. Every hour, 5% of the drug is washed out of his bloodstream. Let dₙ be the amount of drug in Jack’s blood n hours after the drug was taken, where d₀ = 200mg.
72–75. {Use of Tech} Practical sequences
Consider the following situations that generate a sequence
d.Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist.
Drug elimination
Jack took a 200-mg dose of a pain killer at midnight. Every hour, 5% of the drug is washed out of his bloodstream. Let dₙ be the amount of drug in Jack’s blood n hours after the drug was taken, where d₀ = 200mg.
Suppose the sequence {aₙ}⁽∞⁾ₙ₌₀ is defined by the recurrence relation
aₙ₊₁ = ⅓aₙ + 6;a₀ = 3.
b.Explain why {aₙ}⁽∞⁾ₙ₌₀ converges and find the limit.