At the end of year 5, an investment has grown to $750. This investment has earned annual effective rates of interest equal to in= 0.0075- 0.0015 since issue. What was the size of the initial investment made at time 0?

Vaue at the end of year 5 (A5) = Initial Investment *(1+i1)*(1+i2)*(1+i3)*(1+i4)*(1+i5)

i1 = Interest for year1

i2 = Interest for year2

i3 = Interest for year3

i4 = Interest for year4

i5 = Interest for year5

It is given that  i(n)= 0.0075n- 0.0015 , here ''n'' = year

Hence

1+i1 = 1+[0.0075* 1- 0.0015] = 1.006

1+i2 = 1+[0.0075* 2- 0.0015]=1.0135

1+i3 = 1+[0.0075* 3- 0.0015]=1.021

1+i4=  1+[0.0075* 4- 0.0015]=1.0285

1+i5= 1+[0.0075* 5- 0.0015]=1.036

Hence by sustituting the figures we will get,

(A5) = Initial Investment *(1+i1)*(1+i2)*(1+i3)*(1+i4)*(1+i5)

=>$750 = Initial Investment * 1.006* 1.0135 * 1.021 * 1.0285 * 1.036

=>$750 = Initial Investment * 1.109204256

=>Initial Investment = $750 / 1.109204256

=>Initial Investment= $676.16041 (Answer)