question 15 of 20\nat the beginning of year 1, josie invests $400 at an annual compound interest rate of 5%…

question 15 of 20\nat the beginning of year 1, josie invests $400 at an annual compound interest rate of 5%. she makes no deposits to or withdrawals from the account.\nwhich explicit formula can be used to find the accounts balance at the beginning of year 3?\na. a(3)=400·(1 + 0.05)^3 - 1\nb. a(3)=400+(0.05·400)^3 - 1\nc. a(3)=400+(3 - 1)(0.05·400)\nd. a(3)=400·(1 + 0.05)^3

question 15 of 20\nat the beginning of year 1, josie invests $400 at an annual compound interest rate of 5%. she makes no deposits to or withdrawals from the account.\nwhich explicit formula can be used to find the accounts balance at the beginning of year 3?\na. a(3)=400·(1 + 0.05)^3 - 1\nb. a(3)=400+(0.05·400)^3 - 1\nc. a(3)=400+(3 - 1)(0.05·400)\nd. a(3)=400·(1 + 0.05)^3

Answer

Explanation:

Step1: Recall compound - interest formula

The compound - interest formula is $A = P(1 + r)^n$, where $P$ is the principal amount, $r$ is the annual interest rate (in decimal form), and $n$ is the number of years. Here, $P=$400$, $r = 0.05$ (since $5%=0.05$), and we want to find the balance at the beginning of year 3, which means $n = 2$ (because at the beginning of year 1, $n = 0$; at the beginning of year 2, $n = 1$; at the beginning of year 3, $n = 2$). But if we consider the general formula for the $n$th - year balance starting from year 1 as $A(n)$ with $n$ being the number of years passed since the start of the investment, the formula for the balance of the account after $n$ years is $A(n)=P(1 + r)^n$.

Step2: Substitute values

Substitute $P = 400$, $r=0.05$, and $n = 2$ into the formula. However, if we assume the formula is written in terms of the year number starting from 1, for the balance at the beginning of year 3 ($n = 2$ years of compounding have occurred), the formula is $A(3)=400\times(1 + 0.05)^2$. But if we consider the formula in a more general sense where we are just looking at the $n$th - year formula without strict start - of - year accounting in a non - standard way, and we want to find the balance at the "year 3" in the formula's context, the compound - interest formula for the balance $A(n)$ of an account with principal $P$ compounded annually at rate $r$ after $n$ years is $A(n)=P(1 + r)^n$. Here, for $P = 400$, $r=0.05$, and $n = 3$ (counting from the start of the investment as year 1), the formula is $A(3)=400\times(1 + 0.05)^3$.

Answer:

D. $A(3)=400\times(1 + 0.05)^3$