colton is going to invest in an account payin...
colton is going to invest in an account paying an interest rate of 2.4% compounded annually. how much would colton need to invest, to the nearest cent, for the value of the account to reach $4,100 in 7 years?
Answer
# Explanation:
## Step1: Recall compound - interest formula
The compound - interest formula is $A = P(1 + r)^t$, where $A$ is the final amount, $P$ is the principal amount (initial investment), $r$ is the annual interest rate (in decimal form), and $t$ is the number of years. We need to solve for $P$. Rearranging the formula gives $P=\frac{A}{(1 + r)^t}$.
## Step2: Convert the interest rate to decimal
The annual interest rate $r = 2.4\%=0.024$, $A = 4100$, and $t = 7$.
## Step3: Substitute values into the formula
$P=\frac{4100}{(1 + 0.024)^7}$.
First, calculate $(1 + 0.024)^7$. Using the formula $(a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}$, or simply using a calculator, $(1+0.024)^7=1.024^7\approx1.181137$.
Then, $P=\frac{4100}{1.181137}\approx3471.23$.
# Answer:
$3471.23$