Alonzo plans to retire as soon as he has accumulated $250,000 through quarterly payments of $5,000. If Alonzo invests this money at 5.4% interest, compounded quarterly, how long (to the nearest year) until can he retire?

quarterly payment = 5000

i = .054/4 = .0135
n = ?
Amount = 250000

5000(1.0135^n - 1)/.01535 = 250000
1.0135^n - 1 = .0135(50000) = .675
1.0135^n = 1.675
take log of both sides and use log rules
n log 1.0135 = log 1.675
n = 38.47 quarter years or appr 9 years and 7 months

To find out how long it will take for Alonzo to retire, we can use the future value formula for compound interest:

FV = P * (1 + r/n)^(nt)

Where:
FV = Future Value (amount Alonzo wants to accumulate, $250,000)
P = Payment made every quarter ($5,000)
r = Interest rate (5.4%, or 0.054)
n = Number of times the interest is compounded per year (4 times, as there are 4 quarters in a year)
t = Number of years

Plugging in the values we have:

250000 = 5000 * (1 + 0.054/4)^(4t)

Simplifying the equation:

(1 + 0.054/4)^(4t) = 250000/5000
(1 + 0.0135)^(4t) = 50

Now, we can solve for t by taking the logarithm of both sides of the equation:

4t * log(1.0135) = log(50)
t = log(50) / (4 * log(1.0135))

Using a calculator, we find:
t ≈ 19.875

Therefore, it will take Alonzo approximately 19.875 years to retire. Rounded to the nearest year, this is 20 years.

To determine how long it will take Alonzo to retire, we can use the future value of an annuity formula:

A = P * ((1 + r/n)^(nt) - 1) / (r/n)

Where:
A = accumulated value (target amount)
P = payment per period
r = interest rate
n = number of compounding periods per year
t = number of years

In this case, Alonzo plans to accumulate $250,000, with quarterly payments of $5,000. The interest rate is 5.4%, compounded quarterly.

Let's plug in the values into the formula:

250,000 = 5,000 * ((1 + 0.054/4)^(4t) - 1) / (0.054/4)

Simplifying further:

(1 + 0.054/4)^(4t) - 1 = 250,000 * (0.054/4) / 5,000

Now, we can solve for t using logarithms.

(1 + 0.054/4)^(4t) = (250,000 * (0.054/4) / 5,000) + 1

Taking the natural logarithm on both sides:

ln((1 + 0.054/4)^(4t)) = ln((250,000 * (0.054/4) / 5,000) + 1)

Now, we can bring the exponent down:

(4t) * ln(1 + 0.054/4) = ln((250,000 * (0.054/4) / 5,000) + 1)

Finally, solving for t:

t = ln((250,000 * (0.054/4) / 5,000) + 1) / (4 * ln(1 + 0.054/4))

Using a calculator, we can find the value of t to the nearest year.