In each of Exercises 57-60, an initial investment P0, an...

In each of Exercises $57-60,$ an initial investment $P_{0},$ an annual interest rate, and a number $m$ are given. Determine the value of the investment after $m$ months under the following conditions: a. Simple interest b. Semiannual compounding c. Quarterly compounding d. Daily compounding e. Continuous compounding $$ P_{0}=1000, \quad 6 \%, \quad m=12 $$

In each of Exercises $57-60,$ an initial investment $P_{0},$ an annual interest rate, and a number $m$ are given. Determine the value of the investment after $m$ months under the following conditions:
a. Simple interest
b. Semiannual compounding
c. Quarterly compounding
d. Daily compounding
e. Continuous compounding
$$
P_{0}=1000, \quad 6 \%, \quad m=12
$$

Key Concepts

Simple Interest

This concept refers to interest calculated only on the original principal, not on any accumulated interest. Its formula, P = P?(1 + rt), shows that the investment grows linearly over time based solely on the initial amount and the annual rate, without the effect of interest on interest.

Compound Interest

Compound interest involves calculating interest not just on the initial principal but also on the interest that has been previously added. Its formula, P = P?(1 + r/n)^(n*t), reflects how the interest is compounded at regular intervals, causing the investment to grow at an exponential rate as interest accumulates on both the principal and the previously earned interest.

Compounding Frequency

This term describes how often interest is compounded within a given time period, such as semiannually, quarterly, or daily. The frequency of compounding determines the number of times interest is calculated and added to the principal, which affects the final accumulated value of the investment.

Continuous Compounding

Continuous compounding represents the theoretical limit of compound interest, where interest is compounded an infinite number of times per period. The formula P = P?e^(rt) is used to calculate the investment’s value under continuous growth, illustrating the concept of exponential growth when compounding is happening constantly rather than at discrete intervals.

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