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7:3967–70. Formulas for sequences of partial sums Consider the following infinite series.
b.Find a formula for the nth partial sum Sₙ of the infinite series. Use this formula to find the next four partial sums S₅, S₆, S₇, S₈ of the infinite series.
∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]
{Use of Tech} A savings plan
James begins a savings plan in which he deposits \(100 at the beginning of each month into an account that earns 9% interest annually, or equivalently, 0.75% per month.
To be clear, on the first day of each month, the bank adds 0.75% of the current balance as interest, and then James deposits \)100.
Let Bₙ be the balance in the account after the nᵗʰ payment, where B₀ = \$0.
b.Find a recurrence relation that generates the sequence {Bₙ}.
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.
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, 2, 4, 8, 16, ......}
18–20. Evaluating geometric series two ways Evaluate each geometric series two ways.
b. Evaluate the series using Theorem 10.7.
∑ (k = 0 to ∞) (–2/7)ᵏ
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.
