Pre calculus word problem.......?

Question

One thousand dollars is invested at an APR of 4% compounded monthly, which means that .04/12 is the period rate for each month. Find a formula for the value of the investment in terms of months after the initial investment. determine when the investment will reach a value of $2000.

Answer 1

Let P = $1000 (aka, the principal or initial investment)

Let r = 4% / 12 (aka, the period interest rate)

Let n = the period

After the first month (n = 1), the value will be P + P * r.  In other words, we have the original amount (P), plus the interest on that amount (P * r).

So, for n = 1, we can say: Value = P + P * r.  Or, simplifying it algebraically: Value = P * (1 + r)

Now, for n = 1 (the second month), the value is equal to what we had at the end of the first month, plus the interest on that amount.  This is called compound interest.  So for n = 2:

Value (n=2) = [P * (1 + r)] + [P * (1 + r) * r]

Or, simplifying by factoring out the common terms [P * (1 + r)]:

Value(2nd month) = [P * (1 + r)] x (1 + r)

Rewriting that using Value(n=1), which is equal to P * (1 + r), we have:

Value(n=2) = Value(n=1) x (1 + r)

Now it turns out, for the 3rd month, the pattern repeats based on the 2nd month, and you have:

Value(n=3) = Value(n=2) x (1+r)

And so on forever.

Another way to write this, however, is to substitute the original (n=1) equation in, and you end up with:

Value(n) = P * (1+r)(1+r)(1+r)(...etc. up to n)

Writing that mathematically, we have:

Value(n) = P * (1+r)^n

Voila.  There's your formula for the value in month n.

Now, to find in what month that will equal $2000, just plug your numbers in:

$2000 = $1000 * (1 + .04/12) ^ n

Simplify:

2 = (1 + 0.04/12) ^ n

Now you can find the value of n a number of ways.  I'm not sure how far along you are in calculus, but you can use a solving calculator or plug in numbers and zero in on n if you don't know how to use natural logs.  But using natural logs, this equation becomes:

ln(2) = ln[(1 + 0.04/12)^n]

Which simplifies to:

ln(2) = n * ln(1 + 0.04/12)

n = ln(2) / ln(1.0033333)

n = 0.693147181 / 0.00332779

n = 208 months

Answer 2

A=P(1+.0033333333333333)^(48)=2000

p=$1704.74