Respuesta :
The value of a principal amount of $200,000 after 15 years, ar a rate of 8% compounded monthly, is
A = $200,000(1 + 0.08/12)⁽¹² * ¹⁵⁾ = 200000*(1.0067¹⁸⁰) = $661,384.30
If $x is deposited every year, the value of each additional investment will be
x*1.0067¹⁶⁸ = 3.0705x for 14 years
x*1.0067¹⁵⁶ = 2.8341x for 13 years
x*1.0067¹⁴⁴ = 2.6158x for 12 years
and so on
The extra deposits form a geometric sequence with
a = 3.0705x
r = 0.923
The sum of the 14 terms of the sequence is
[tex] \frac{a(1-r^{14})}{1-r} = \frac{3.0705x(1-0.923^{14})}{1-0.923} =26.8886x[/tex]
The total investment is worth $1,000,000 after 15 years. Therefore
661,384.30 + 26.8886x = 1,000,000
26.8886x = 338,615.70
x = $12,593.28
Answer:
James saves $12,593.28 each year.
A = $200,000(1 + 0.08/12)⁽¹² * ¹⁵⁾ = 200000*(1.0067¹⁸⁰) = $661,384.30
If $x is deposited every year, the value of each additional investment will be
x*1.0067¹⁶⁸ = 3.0705x for 14 years
x*1.0067¹⁵⁶ = 2.8341x for 13 years
x*1.0067¹⁴⁴ = 2.6158x for 12 years
and so on
The extra deposits form a geometric sequence with
a = 3.0705x
r = 0.923
The sum of the 14 terms of the sequence is
[tex] \frac{a(1-r^{14})}{1-r} = \frac{3.0705x(1-0.923^{14})}{1-0.923} =26.8886x[/tex]
The total investment is worth $1,000,000 after 15 years. Therefore
661,384.30 + 26.8886x = 1,000,000
26.8886x = 338,615.70
x = $12,593.28
Answer:
James saves $12,593.28 each year.
