Question 1115943
your investment in the shares in year 1 is equal to 1000.


the formula for your investment in the shares in year n is equal to 1000 * (1.03) ^ (n-1)


this is because you are increasing the amount you invest each year by 3% per year.


in year 1, it is 1000 * 1.03 ^ 0 = 1000


in year 2, it is 1000 * 1.03 ^ 1 = 1030


in year 3, it is 1000 * 1.03 ^ 2 = 1060.9


etc.....


this can be seen in column C of the excel spreadsheet printout.


the shares cost 50 dollars each in year 1.
they increase in value by 12% a year.


the formula for the price of each share in year n is 50 * (1.12) ^ (n-1)


in year 1, they cost 50 * 1.12 ^ 0 = 50


in year 2, they cost 50 * 1.12 ^ 1 = 56


in year 3, they cost 50 * 1.12 ^ 2 = 62.72


etc.....


this can be seen in column D of the excel spreadsheet printout.


the formula for the number of shares bought in year n is (1000 * (1.03) ^ (n-1)) / (50 * (1.12) ^ (n-1))


the number of shares bought in year 1 is (1000 * 1.03 ^ 0) / (50 * 1.12 ^ 0) = 20

the number of shares bought in year 2 is (1000 * 1.03 ^ 1) / (50 * 1.12 ^ 1) = 18.39285714.


the number of shares bought in year 3 is (1000 * 1.03 ^ 2) / (50 * 1.12 ^ 2) = 16.91485969.


etc.....


this can be seen in column E of the excel spreadsheet printout.


the number of shares bought each year is being reduced because your investment is increasing 3% per year and the price of each share has been increasing 12% per year.


the number of shares you buy each year is therefore equal to (3000 * 1.03 ^ (n-1)) / (50 * 1.12 ^ (n-1)


this formula can be shown as (3000 / 50) * (1.03 ^(n-1)) / 1.12 ^ (n-1)).


this can also be shown as (3000 / 50) * (1.03/1.12) ^ (n-1)


simplify this to get the number of shares that can be bought in year n is 20 * .9196428571 ^ (n-1)


for example, the number of shares that can be bought in year 9 is equal to 50 * .9196428571 ^ (8) which is equal to 10.23254379.


look at cell E11 in the excel spreadsheet printout and you will see that the number of shares bought in year 9 is equal to 10.23254379.


cell E11 is the cell that is in column E row 11.


as you can see in the excel spreadsheet printout, the number of shares bought in each succeeding year is less even though the amount of money invested in each year is more because the price of each share is increasing at a greater rate than the amount being invested.


here's the excel spreadsheet printout.


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here's a reference that can help you understand how mutual shares work.


<a href= "https://www.thebalance.com/buying-mutual-funds-is-different-than-buying-stocks-3140909" target = "_blank">https://www.thebalance.com/buying-mutual-funds-is-different-than-buying-stocks-3140909</a>