Question

1. sara starts a new job and deposits the same paycheck in her checking account each month. after 3 months, her checking account balance is $2,500. it increases to $3,100 after four months and $3,700 after five months. assume she makes no withdrawals. create a table to represent her account balance: (no symbols like $, just the number)

x (months)
y (balances$)
0
1
2
3
4
5

Explanation:

Step1: Find the monthly deposit

We know that from month 3 to month 4, the balance increases from 2500 to 3100? Wait, no, wait. Wait, the problem says "After 3 months, her checking account balance is $2,500. It increases to $3,100 after four months and $3,700 after five months." Wait, so from month 3 to month 4, the increase is 3100 - 2500 = 600? Wait, no, that can't be. Wait, maybe the initial deposit? Wait, no, let's re - read. "Sara starts a new job and deposits the same paycheck in her checking account each month. After 3 months, her checking account balance is $2,500. It increases to $3,100 after four months and $3,700 after five months." Wait, so the difference between month 3 and month 4 is 3100 - 2500 = 600, and between month 4 and month 5 is 3700 - 3100 = 600. So the monthly deposit (the common difference) is 600. Now, let's find the balance at month 0, 1, 2.

Let's assume the balance at month \(x\) is \(y = mx + b\), where \(m\) is the slope (monthly deposit) and \(b\) is the initial balance (at \(x = 0\)). We know that at \(x = 3\), \(y = 2500\); at \(x = 4\), \(y = 3100\). The slope \(m=\frac{3100 - 2500}{4 - 3}=\frac{600}{1}=600\).

Now, use the point - slope form \(y - y_1=m(x - x_1)\). Using the point \((3,2500)\):

\(y-2500 = 600(x - 3)\)

\(y=600x-1800 + 2500\)

\(y = 600x+700\)

Now, find the balance for \(x = 0\): \(y=600(0)+700 = 700\)

For \(x = 1\): \(y=600(1)+700=1300\)

For \(x = 2\): \(y=600(2)+700 = 1200 + 700=1900\)

For \(x = 3\): 2500 (given)

For \(x = 4\): 3100 (given)

For \(x = 5\): 3700 (given)

Step2: Create the table

x (Months)	0	1	2	3	4	5

Answer:

The table is as follows:

x (Months)	0	1	2	3	4	5

So the values for the boxes (from \(x = 0\) to \(x = 5\)) are 700, 1300, 1900, 2500, 3100, 3700 respectively.