Solve each problem. See Examples 6– 8. How much money will...

Solve each problem. See Examples 6– 8.
How much money will be in an account at the end of 8 yr if $\$ 4500$ is deposited at $6 \%$ annual interest compounded as follows?
(a) Annually
(b) Semiannually
(c) Quarterly
(d) Daily (Use $n=365 .$ )
(e) Continuously

Key Concepts

Compound Interest

Compound interest is the accumulation of interest on both the initial principal and the accumulated interest from previous periods. It is calculated using a formula that takes into account the initial deposit, the periodic interest rate, and the number of compounding periods. This concept is fundamental in finance as it illustrates how investments can grow exponentially over time when interest is reinvested rather than withdrawn.

Compounding Frequency

Compounding frequency refers to the number of times per year that interest is calculated and added to the principal. Common frequencies include annually, semiannually, quarterly, and daily. The more frequently interest is compounded, the more interest is accrued overall, because interest is being calculated and added to the balance more often. This concept is key to understanding how small differences in compounding intervals can significantly affect the final amount deposited.

Continuous Compounding

Continuous compounding occurs when interest is calculated and added an infinite number of times in a given period, leading to the maximum possible amount of interest accrued. Instead of using a discrete compounding period, continuous compounding employs the mathematical constant e through the formula A = Pe^(rt), where r is the interest rate and t is time. This concept is important in finance and mathematics as it models the most idealized form of growth.

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