Question

You have 42,180.53 in a brokerage account, and you plan to deposit and additional 5,000 at the end of every future year until your account total 250,000. you expect to earn 12% annually on the account. How many years will it take to reach your goal?

Answer 1

Let "x" be the number of years.
(42180.53+5000x)(1.12x)=250000
5600x^2+47242.1936x-250,000=0
I graphed this and found x=3.68 years
You might want to round that out to 4 yrs

Or

The $5,000 is considered an annuity for which we want to find the FV; while the $42,180.53 is a lump sum for which we also want to find the future value after n number of years.
Therefore, the sum of the FV of the annuity & the future value of the lump sum should be $250,000 after n number of years.
To find the FV of an annuity:

FV= c*((1+r)^n-1)/r
= 5000*(1.12^n-1)/0.12
To find the FV of a lump sum:

FV= c*(1+r)^n
= 42180.53*(1.12)^n
Now:
$250,000 = 5000*(1.12^n-1)/0.12 + 42180.53*(1.12)^n
Since calculations are complicated, I have used excel to answer the problem, but you can use a financial calculator as follows:
Using your financial calculator, enter the following data: I = 12;
PV = -42180.53; PMT = -5000; FV = 250000; N = ? Solve for N = 11. It will take
11 years for John to accumulate $250,000.