College

A store had 150 laptops in the month of January. Every month, 20% of the laptops were sold, and 10 new laptops were stocked in the store.

Which recursive function best represents the number of laptops in the store [tex]f(n)[/tex] after [tex]n[/tex] months?

A. [tex]f(n) = 150 - 0.8 \cdot f(n-1) + 10, \, f(0) = 150, \, n > 0[/tex]

B. [tex]f(n) = 0.2 \cdot f(n-1) + 10, \, f(0) = 150, \, n > 0[/tex]

C. [tex]f(n) = 0.8 \cdot f(n-1) + 10, \, f(0) = 150, \, n > 0[/tex]

D. [tex]f(n) = 150 + 0.8 \cdot f(n-1) + 10, \, f(0) = 150, \, n > 0[/tex]

Answer :

To solve this problem, we're asked to determine which recursive function best represents the number of laptops in the store, denoted by [tex]\( f(n) \)[/tex], after [tex]\( n \)[/tex] months.

Here's a step-by-step breakdown of how to interpret the problem and find the correct recursive function:

1. Initial Condition:
- The store starts with 150 laptops in January, so [tex]\( f(0) = 150 \)[/tex].

2. Monthly Laptop Sales:
- Every month, 20% of the existing laptops are sold. This means that 80% of the laptops remain in the store each month. If [tex]\( f(n-1) \)[/tex] represents the number of laptops at the start of month [tex]\( n \)[/tex], then the number of laptops remaining after sales is [tex]\( 0.8 \cdot f(n-1) \)[/tex].

3. Restocking:
- Each month, 10 new laptops are added to the store. Therefore, after restocking, the number of laptops becomes [tex]\( 0.8 \cdot f(n-1) + 10 \)[/tex].

4. Recursive Function:
- Combining these observations, the recursive function that represents the situation is:
[tex]\[
f(n) = 0.8 \cdot f(n-1) + 10
\][/tex]
- This captures the effect of both selling some laptops and adding new ones each month.

5. Initial Condition with Recursive Formula:
- We start with [tex]\( f(0) = 150 \)[/tex], and for each subsequent month, [tex]\( f(n) = 0.8 \cdot f(n-1) + 10 \)[/tex].

Based on these steps, the function that correctly describes the number of laptops in the store after [tex]\( n \)[/tex] months is:

[tex]\( f(n) = 0.8 \cdot f(n-1) + 10 \)[/tex], with [tex]\( f(0) = 150 \)[/tex].