here are two ways of investing $50,000 for 25...
here are two ways of investing $50,000 for 25 years. lump - sum deposit rate time $50,000 4% compounded annually 25 years periodic deposit rate time $2000 at the end of each year 4% compounded annually 25 years use this information and the formulas a = p(1 + r)^t and a = p(1 + r)^t - 1/r to complete parts a. and b. below. a. after 25 years, how much more will you have from the lump - sum investment than from the annuity? you will have approximately $ more from the lump - sum investment than from the annuity. (round to the nearest dollar as needed.)
Answer
# Explanation:
## Step1: Calculate lump - sum amount
Use the formula $A = P(1 + r)^t$, where $P=\$50000$, $r = 0.04$, and $t = 25$.
$A_{lump - sum}=50000\times(1 + 0.04)^{25}$
$A_{lump - sum}=50000\times2.6658377$
$A_{lump - sum}=1332918.85$
## Step2: Calculate annuity amount
Use the formula $A=\frac{P[(1 + r)^t-1]}{r}$, where $P = 2000$, $r=0.04$, and $t = 25$.
$A_{annuity}=\frac{2000\times[(1 + 0.04)^{25}-1]}{0.04}$
First, calculate $(1 + 0.04)^{25}-1=2.6658377 - 1=1.6658377$.
Then $A_{annuity}=\frac{2000\times1.6658377}{0.04}$
$A_{annuity}=2000\times41.6459425$
$A_{annuity}=83291.885$
## Step3: Find the difference
Subtract the annuity amount from the lump - sum amount.
$Difference=A_{lump - sum}-A_{annuity}$
$Difference = 1332918.85-83291.885$
$Difference=1249626.965\approx1249627$
# Answer:
$1249627$