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Select the correct answer.

Each month Barry makes three transactions in his checking account:
- He deposits [tex]\$700[/tex] from his paycheck.
- He withdraws [tex]\$150[/tex] to buy gas for his car.
- He withdraws [tex]\$400[/tex] for other expenses.

His account balance is [tex]\$1,900[/tex] at the end of the 1st month. Which recursive equation models Barry's account balance?

A. [tex]f(1) = 1,900[/tex]
[tex]f(n) = f(n-1) + 150, \text{ for } n \geq 2[/tex]

B. [tex]f(1) = 1,800[/tex]
[tex]f(n) = f(n-1) + 700, \text{ for } n \geq 2[/tex]

C. [tex]f(1) = 1,900[/tex]
[tex]f(n) = 150 \cdot f(n-1), \text{ for } n \geq 2[/tex]

D. [tex]f(1) = 1,900[/tex]
[tex]f(n) = f(n-1) - 150, \text{ for } n \geq 2[/tex]

Answer :

Let's work through the problem step by step to find the correct recursive equation that models Barry's account balance.

1. Understanding Barry's transactions:
- Each month, Barry deposits [tex]$700 from his paycheck into his checking account.
- He withdraws $[/tex]150 for gas.
- He also withdraws [tex]$400 for other expenses.
- The total amount withdrawn each month is $[/tex]150 + [tex]$400 = $[/tex]550.

2. Calculate the net change in the account per month:
- Deposit: [tex]$700
- Total withdrawals: $[/tex]550
- Net change per month = Deposit - Withdrawals = [tex]$700 - $[/tex]550 = [tex]$150

3. Initial condition:
- At the end of the first month, Barry's account balance is $[/tex]1,900.

4. Modeling the account balance with a recursive equation:
- We are given several options to choose from. We want to find the equation that correctly represents the situation.

Let's evaluate the relevant option:
- Option A. [tex]\( f(1) = 1,900 \)[/tex]
- Recursive formula: [tex]\( f(n) = f(n-1) + 150 \)[/tex], for [tex]\( n \geq 2 \)[/tex]

This option starts with an initial balance of [tex]$1,900, which matches Barry's balance at the end of the first month. The recursive formula adds $[/tex]150 per month, which accurately reflects the net change in Barry's account.

Therefore, the correct recursive equation that models Barry's account balance is option A.

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