Final answer:
To calculate the growth of a $420 investment that doubles every 7 years after 16 years, find the full doubling periods and the growth for the remaining time, which leads to approximately $2094 in the account after 16 years.
Step-by-step explanation:
Calculation of Investment Growth
The amount of money in an account after a certain number of years can be calculated using the formula for compound interest. As the money doubles every 7 years, this represents an exponential growth according to the rule of 72. To find the total after 16 years, we divide 16 by 7 to calculate the number of full doubling periods, which in this case is about 2.29 periods. Since the investment is not exactly at a doubling point at the 16-year mark, we can use the power of compounding to estimate the final amount.
For the first 14 years (or two doubling periods), the $420 investment will double twice:
After 7 years: $420 × 2 = $840
After 14 years: $840 × 2 = $1680
For the remaining 2 years, we need to find the growth factor for a 7-year period (which is 2) and then raise it to the fraction of time (2/7 years) that we're interested in. This can be represented as:
$1680 × 2^(2/7) = $1680 × approximately 1.246 = $2094
Rounding this to the nearest dollar, we have $$2094 in the account after 16 years.