The formula for the amount A in an investment account with a nominal interest rate r at any time t is given by A(t) = a(e)rt, where a is the

The formula for the amount A in an investment account with a nominal interest rate r at any time t is given by A(t) = a(e)^rt, where a is the amount of principal initially deposited into an account that compounds continuously. Prove that the percentage of interest earned to principal at any time t can be calculated with the formula I(t) = e^rt − 1.

Expert Solution

Consider the formula for nominal interest as following:

A(t) = a(e)rt

The annual percentage is determined using the formula:

I(t) = ert - 1

Simplify the above equation as follows:

I(t) + 1 = ert – 1 + 1

I(t) + 1 = ert

Now take natural log on both sides of the above equation,

ln{I(t) + 1} = ln(ert)

rt = ln{I(t) + 1}

r = ln{I(t) + 1}/t

Therefore, the percentage of interest is:

r = ln{I(t) + 1}/t.

Therefore, the percentage of interest is:

r = ln{I(t) + 1}/t.

Nabeel answered 1 year ago

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