Question

at the time of her grandsons birth, a grandmother deposits $5000 in an account that pays 2.5% compounded monthly. what will be the value of the account at the childs twenty - first birthday, assuming that no other deposits or withdrawals are made during this period? click the icon to view some finance formulas. the value of the account will be $\square$. (round to the nearest dollar as needed)

Explanation:

Step1: Recall compound interest formula

The compound - interest formula for compounding $n$ times a year is $A = P(1+\frac{r}{n})^{nt}$, where:

• $A$ is the amount of money accumulated after $n$ years, including interest.
• $P$ is the principal amount (the initial amount of money).
• $r$ is the annual interest rate (in decimal form).
• $n$ is the number of times that interest is compounded per year.
• $t$ is the time the money is invested for in years.

Given:

• $P=\$5000$
• $r = 2.5\%=0.025$
• $n = 12$ (since it is compounded monthly)
• $t = 21$ years (from the birth of the grandson to his 21st birthday)

Step2: Substitute the values into the formula

Substitute $P = 5000$, $r=0.025$, $n = 12$ and $t = 21$ into the formula $A=P(1 +\frac{r}{n})^{nt}$:

First, calculate $\frac{r}{n}=\frac{0.025}{12}\approx0.00208333$

Then, calculate $nt=12\times21 = 252$

Next, calculate $1+\frac{r}{n}=1 + 0.00208333=1.00208333$

Then, calculate $(1+\frac{r}{n})^{nt}=(1.00208333)^{252}$

Using a calculator, $(1.00208333)^{252}\approx1.64755$

Finally, calculate $A=5000\times1.64755 = 5000\times(1+\frac{0.025}{12})^{12\times21}$

$A=5000\times(1.00208333)^{252}\approx5000\times1.64755 = 8237.75\approx8238$ (rounded to the nearest dollar)

Answer:

$\$8238$