You are trying to decide how much to save for retirement. Assume you plan to save $ 6,500 per year with the first investment made one year from now. You think you can earn 6.5% per year on your investments and you plan to retire in 42 years, immediately after making your last $6,500 investment. Required:
a. How much will you have in your retirement account on the day you retire?
b. If, instead of investing $ 6,500 per year, you wanted to make one lump-sum investment today for your retirement that will result in the same retirement saving, how much would that lump sum need to be?
c. If you hope to live for 26 years in retirement, how much can you withdraw every year in retirement (starting one year after retirement) so that you will just exhaust your savings with the 26th withdrawal (assume your savings will continue to earn 11.0% in retirement)?

a. How much will you have in your retirement account on the day you retire?

we must calculate the future value of an annuity:

future value = annuity payment x {[(1 + r)ⁿ - 1] / r} = $6,500 x {[(1 + 0.065)⁴² - 1] / 0.065} = $1,308,262.21

b. If, instead of investing $ 6,500 per year, you wanted to make one lump-sum investment today for your retirement that will result in the same retirement saving, how much would that lump sum need to be?

future value of lump sum = lump sum x (1 + r)ⁿ

$1,308,262.21 = lump sum x (1 + 0.065)⁴²

lump sum = $1,308,262.21 / (1 + 0.065)⁴² = $1,308,262.21 / 14.08262214 = $92,899.05

c. If you hope to live for 26 years in retirement, how much can you withdraw every year in retirement (starting one year after retirement) so that you will just exhaust your savings with the 26th withdrawal (assume your savings will continue to earn 11.0% in retirement)?

present value of an annuity = annuity payment x {[1 - 1/(1 + r)ⁿ ] / r}

annuity payment = $1,308,262.21 / {[1 - 1/(1 + r)ⁿ ] / r} = $1,308,262.21 / {[1 - 1/(1 + 0.11)²⁶ ] / 0.11} = $1,308,262.21 / 8.48806 = $154,129.74

start by isolating the variable on one side by adding 43w^2 to both sides.   the equation will read:

25w^2+5=0

*subtract 5 from both sides*

25w^2=-5

*divide both sides by 25*

w^2=-1/5

*take the square root of both sides*

w=√(-1/5)

the square root of -1 is i.   the square root of 5 remains √5

the answer is i/√5.

hope this