Question

When interest is compounded n times a year, the accumulated amount (A) after t years is given by the formula

A = P(1+r/n)^nt

where P is the initial principal and r is the annual rate of interest. Approximately how long will it take $2,000 to double at an annual interest rate of 5.25% compounded monthly?

Answer 1

To find the approximate time it takes for $2,000 to double at an annual interest rate of 5.25% compounded monthly, we can use the formula for compound interest, which is A = P(1+r/n)^nt. Substituting the given values into the formula and solving the resulting equation, we can find the value of t.

Step-by-step explanation:

To find the approximate time it takes for $2,000 to double at an annual interest rate of 5.25% compounded monthly, we can use the formula for compound interest, which is A = P(1+r/n)^nt.

In this case, the initial principal (P) is $2,000, the annual interest rate (r) is 5.25% (or 0.0525), and it is compounded monthly, so n = 12. We need to find the time (t) when the accumulated amount (A) is twice the initial principal.

Let's substitute these values into the formula: A = $2,000 * (1 + 0.0525/12)^(12t). We want A to be equal to $4,000, so the equation becomes $4,000 = $2,000 * (1 + 0.0525/12)^(12t).

We can solve this equation to find the value of t. First, divide both sides of the equation by $2,000: 2 = (1 + 0.0525/12)^(12t). Take the natural logarithm (ln) of both sides to isolate the exponent: ln(2) = ln((1 + 0.0525/12)^(12t)). Use the logarithmic identity to bring down the exponent: ln(2) = 12t * ln(1 + 0.0525/12). Finally, divide both sides by 12 * ln(1 + 0.0525/12) to solve for t: t = ln(2) / (12 * ln(1 + 0.0525/12)). Use a calculator to find the approximate value of t.