Suppose you pay $1000 into a savings account that pays 2% per year, compounded annually. How many years will would it take for the money in the account to double, to $2000? use "successive approximation."

P = Po(1+r)^n.

P = 1000(1.02)^n = 2000
1.02^n = 2
n*Log 1.02 = Log 2
n = Log 2/Log 1.02 = 35 Compounding periods.

35Comp / 1Comp/yr = 35 Yrs. To double.

To find out how many years it would take for the money in the account to double using successive approximation, you can use a formula for compound interest.

The formula for compound interest can be written as:

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

Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (in this case, $1000)
r = annual interest rate (in this case, 2%, which can be written as 0.02)
n = number of times that interest is compounded per year (in this case, once annually)
t = number of years the money is invested (the unknown we are trying to find)

We know that we want the investment to double, so we can set A to be $2000. Now we have the equation to solve:

2000 = 1000(1 + 0.02/1)^(1 * t)

To solve this equation using successive approximation, we can start with an initial estimate for t and refine it until we find the answer.

Let's start with an initial estimate of t = 5 years. Plug this value into the equation:

2000 = 1000(1 + 0.02)^(5)

Calculate the right side of the equation:

2000 = 1000(1.02)^(5)
2000 = 1000(1.10408)

Divide both sides by 1000 to isolate the base:

2 = 1.10408

The left side of the equation is smaller than the right side, indicating that our initial estimate of 5 years is too low. We need to increase our estimate.

Try a new estimate of t = 6 years:

2000 = 1000(1 + 0.02)^(6)
2000 = 1000(1.02)^(6)
2000 = 1000(1.1259)

Again, divide both sides by 1000:

2 = 1.1259

The left side is still smaller than the right side, so we need to increase our estimate again.

Try a new estimate of t = 7 years:

2000 = 1000(1 + 0.02)^(7)
2000 = 1000(1.02)^(7)
2000 = 1000(1.149)

Divide both sides by 1000:

2 = 1.1487

This time, the left side is larger than the right side, indicating that our estimate of 7 years is too high. We need to refine our estimate.

Continue this process of refining the estimate by trying different values until you narrow down a range where the left side is very close to the right side (e.g., 1.99995).

In this case, you would continue with t = 7.5, 7.6, 7.7, and so on, until you find the value of t where the left side is nearly equal to the right side. In this example, the result is approximately 7.27 years.

Therefore, it would take approximately 7.27 years for the money in the account to double to $2000.