What is my final amount of money after.....(maths)?

Question

If I initially invest $5000 into a bank account at 8.6% per annum and then $50 a week for 20 years, how much money will I have at the end of the 20 years? Please also show your working if possible :)

Answer 1

$ 5000 + 0.086($5000)(20) + 20(52)(50) = x

5000 + 430(20) + 52000 = x

5000 + 8600 + 52000 = x

x = $ 65,600

*1 yr = 52 weeks - this is where i got the 52

Answer 2

Assume that the interest is compounded weekly and that you added you first $50 one week after you invested your $5000 and keep on investing $50 every week for  20 years.

Let:

S = the amount at the end of the periods

n = number of periods = (20 x365 + 5)/7 +1 = 1044 weeks

i = interest per period = 0.086/52.14286

1. S = 5000(1+i)^n + 50(1 + i)^(n-1) +.............. 50(1+i)

Multiplying Eq.1 by(1 + i):

2. S(1 + i) = 5000((1 + i)^n)(1 + i) + 50(1 + i)^n ......50(1 + i)^2

Subtracting Eq.1 from Eq.2:

3. S(i) = 5000(1 + i)^n(i) + 50(1 + i)^n - 50(1 + 1)

4. S = 5000(1 + i)^n +(50(1 + i)((1 + i)^(n-1) - 1)/i

S = 5000(1 + 0.086/52.14286)^1044 + 50(1 + 0.086/52.14286)((1 + 0.086/52.14286)^1043 -1)/(0.08/52.14286) = $166947.44