Bethany starts investing #300 per month at 6% per annum, compounded monthly, for 5 years. After 3 years, the interest increases to 9% annum, compounded monthly. Determine the amount of her investment after 5 years.

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There are two separate calculations to this question.  We need to first determine the amount of the investment for the first three years, then use that result to find the amount of the investment for the remaining two years.

Compound interest investments for annuities are determined using the formula

`A=R{(1+i)^n-1}/i`  where A is the amount of the investment, R is the rent (or monthly payment), i is the interest rate per period and n is the number of investment periods.

For the first 3 years, we have `3 times 12=36` payments, each at $300 and an interest rate of 6%/a = 0.06/12=0.005.  This means the amount at the end of 3 years is

`A=(300)(1.005^36-1)/0.005 =11800.83`

Now this amount will grow at 9% per year for 2 more years, so its final amount will be:

`A=11800.83(1.09)^2 approx 14020.57`

In addition, we have the $300 per month into the annuity but now `i=0.09/12=0.0075` and `n=2 times 12=24` .  This annuity is

`A=300(1.0075^24-1)/0.0075 approx 7856.54`

The total amount of the investment is the two annuities added together to get `14020.57+7856.54 = 21877.11` .

The investment is worth $21877.11 after 5 years.


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