Question

A financial advisor tells you that you can make your child a millionaire if you just start saving early. You decide to put an equal amount each year into an investment account that earns 7.5% interest per year, starting on the day your child is born. How much would you need to invest each year (rounded to the nearest dollar) to accumulate a million for your child by the time he is 35 years old? (Your last deposit will be made on his 34th birthday.)​

Answer 1

$12159 per year.

Explanation:

If I invest $x each year at the simple interest of 7.5%, then the first $x will grow for 35 years, the second $x will grow for 34 years and so on.

So, the total amount that will grow after 35 years by investing $x at the start of each year at the rate of 7.5% simple interest will be given by

`x( 1 + (35 * 7.5)/(100)) + x( 1 + (34 * 7.5)/(100)) + x( 1 + (33 * 7.5)/(100)) + ......... + x( 1 + (1 * 7.5)/(100))`

=
`35x + (x * 7.5)/(100) [35 + 34 + 33 + ......... + 1]`

=
`35x + (x * 7.5)/(100) [(1)/(2) (35) (35 + 1)]`

{Since sum of n natural numbers is given by
`(1)/(2) (n)(n + 1)`}

= 35x + 47.25x

= 82.25x

Now, given that the final amount will be i million dollars = $1000000

So, 82.25x = 1000000

⇒ x = $12,158. 05 ≈ $12159

Therefore. I have to invest $12159 per year. (Answer)