If 1000 is invested at 6 % interest, compounded annually,...

If $$\$ 1000$$ is invested at $6 \%$ interest, compounded annually, then after $n$ years the investment is worth $a_n=1000(1.06)^n$ dollars.
(a) Find the first five terms of the sequence $\left\{a_n\right\}$.
(b) Is the sequence convergent or divergent? Explain.

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Key Concepts

Convergence and Divergence of Sequences

Convergence and divergence are properties that describe the long-term behavior of sequences. A sequence converges if its terms approach a finite limit as n becomes very large, and it diverges if the terms do not approach a finite value. In this problem, because the base of the exponential (1.06) is greater than 1, the sequence grows without bound as n increases, which means the sequence diverges.

Exponential Growth

Exponential growth refers to a pattern where a quantity increases by a constant multiplicative factor over equal increments of time. In the context of the problem, the investment of $1000 growing at 6% per year is modeled by the exponential expression 1000(1.06)^n. Each term in the sequence is obtained by multiplying the previous term by 1.06, illustrating the fundamental property of exponential growth.

Compound Interest

Compound interest is the process where interest is earned on both the initial principal and on the accumulated interest from previous periods. The formula used here, a_n = 1000(1.06)^n, comes from the compound interest formula, which mathematically describes how an investment grows over time when interest is compounded annually. This concept is central to many real-world financial calculations.

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