Understanding e and Compound Interest

The video explores the mathematical constant e through the lens of compound interest. It begins with an introduction to e and its natural occurrence in finance, followed by a demonstration of how values converge to e using Excel. The instructor then generalizes interest rates with a variable r and explains the calculations involved in compounding interest over time. Finally, the concept of continuous compounding is introduced, showing how it relates to the limit of (1 + 1/n)^n as n approaches infinity, ultimately connecting to the constant e.

What is the significance of the number 2.7183 in the context of compound interest?

• It is the number of compounding periods in a year.

• It is the base of the natural logarithm.

• It is the principal amount in the formula.

• It is the maximum interest rate possible.

Which of the following statements about the number e is true?

• e is only used in finance.

• e is unrelated to pi.

• e is a rational number.

• e appears in various mathematical contexts.

How is the interest rate represented in the generalized compound interest formula?

• As a percentage directly.

• As a whole number.

• As a decimal value.

• As a fraction of the principal.

If the interest rate is 25%, how would it be represented in the formula?

• 0.25

• 25

• 0.025

• 2.5

In the example given, what is the compounding frequency if n equals 2?

• Semi-annually

• Annually

• Monthly

• Quarterly

What happens to the amount owed as the number of compounding periods increases indefinitely?

• It becomes infinite.

• It converges to a value involving e.

• It remains constant.

• It decreases to zero.

What substitution is made to find the limit as n approaches infinity in continuous compounding?

• 1 over x equals n over r

• x equals r over n

• x equals n times r

• 1 over x equals r over n

What is the result of continuously compounding an interest rate of 10% over one year?

• 0.1 times the principal

• e to the power of 0.1 times the principal

• 10 times the principal

• e times the principal

What is the relationship between n and x in the context of continuous compounding?

• n equals x times r

• n equals r over x

• n equals x over r

• n equals r times x

What does the expression 1 plus 1 over x to the x power approach as x approaches infinity?

• Zero

• One

• e

• Infinity