Question

lucas deposits $2,000 in an account that earns simple interest. kayla deposits $2,000 in an account that earns compound interest. the table shows the balances of the accounts after different numbers of years.

time (years)

lucass balance (dollars)

kaylas balance (dollars)

----

----

----

0

2,000

2,000

1

2,100

2,100

2

2,200

2,205

5

2,500

2,552.56

8

2,800

2,954.91

which function represents kaylas account balance after $x$ years?

Explanation:

Step1: Identify compound interest formula

The compound interest formula is $A = P(1 + r)^x$, where $P$ is principal, $r$ is annual interest rate, $x$ is time in years, $A$ is the balance.

Step2: Use year 1 data to find $r$

Substitute $P=2000$, $x=1$, $A=2100$:
$2100 = 2000(1 + r)^1$
$\frac{2100}{2000} = 1 + r$
$1.05 = 1 + r$
$r = 0.05$

Step3: Verify rate with year 2 data

Substitute $P=2000$, $x=2$, $r=0.05$:
$A = 2000(1 + 0.05)^2 = 2000(1.1025) = 2205$, which matches the table.

Step4: Finalize the function

Substitute $P=2000$ and $r=0.05$ into the formula:
$A = 2000(1.05)^x$

Answer:

$A(x) = 2000(1.05)^x$ (where $A(x)$ is Kayla's balance after $x$ years)