Question

if $2000 is invested at 6% simple interest, the initial value and the future value s at the end of each of 5 years is shown in the table below. then answer part (a) through (d).

year (t)	future value ($)
1	2120
2	2240
3	2360
4	2480
5	2600

(a) can a linear function model exactly the points from the table? explain.
a. no, the first differences are not constant for uniform inputs.
b. yes, the first differences are constant for uniform inputs.
(b) if so, find a linear function $s = f(t)$ that models the points. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. \\(\boxed{\text{(type an equation.)}}\\)
b. a linear model cannot exactly model the points from the table.

Explanation:

Step1: Check first differences

Calculate differences in $S$:
$2120-2000=120$, $2240-2120=120$, $2360-2240=120$, $2480-2360=120$, $2600-2480=120$
Input $t$ increases by 1 uniformly.

Step2: Determine linear model eligibility

Constant first differences mean linear fit.

Step3: Find linear equation

Slope $m=120$, y-intercept $b=2000$.
Linear form: $S=mt+b$
Substitute values: $S=120t+2000$

Answer:

(a) B. Yes, the first differences are constant for uniform inputs.
(b) A. $S=120t+2000$