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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].
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].
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