Answer :
To solve this problem, we need to determine the value of [tex]\( F(101) \)[/tex] based on the information given.
We know two things:
1. [tex]\( F(1) = 2 \)[/tex].
2. The function is defined recursively: [tex]\( F(n) = F(n-1) + \frac{1}{2} \)[/tex] for all integer [tex]\( n \geq 1 \)[/tex].
Our goal is to find [tex]\( F(101) \)[/tex].
Let's break it down step-by-step:
1. Start with the initial condition: We know that [tex]\( F(1) = 2 \)[/tex].
2. Use the recursive formula to find subsequent terms:
- For [tex]\( F(2) \)[/tex], use [tex]\( F(2) = F(1) + \frac{1}{2} = 2 + \frac{1}{2} = 2.5 \)[/tex].
- For [tex]\( F(3) \)[/tex], use [tex]\( F(3) = F(2) + \frac{1}{2} = 2.5 + \frac{1}{2} = 3.0 \)[/tex].
3. Notice the pattern: Each time we move to the next term, we add [tex]\(\frac{1}{2}\)[/tex].
4. Calculate [tex]\( F(101) \)[/tex]:
- Since we start at 2 for [tex]\( F(1) \)[/tex], and from [tex]\( F(2) \)[/tex] onward we add [tex]\(\frac{1}{2}\)[/tex] each step, the amount added to reach [tex]\( F(101) \)[/tex] will be [tex]\( \frac{1}{2} \times (101-1) = \frac{1}{2} \times 100 = 50 \)[/tex].
5. Find the final value:
- So, [tex]\( F(101) = F(1) + 50 = 2 + 50 = 52 \)[/tex].
The correct answer is [tex]\( C. 52 \)[/tex].
We know two things:
1. [tex]\( F(1) = 2 \)[/tex].
2. The function is defined recursively: [tex]\( F(n) = F(n-1) + \frac{1}{2} \)[/tex] for all integer [tex]\( n \geq 1 \)[/tex].
Our goal is to find [tex]\( F(101) \)[/tex].
Let's break it down step-by-step:
1. Start with the initial condition: We know that [tex]\( F(1) = 2 \)[/tex].
2. Use the recursive formula to find subsequent terms:
- For [tex]\( F(2) \)[/tex], use [tex]\( F(2) = F(1) + \frac{1}{2} = 2 + \frac{1}{2} = 2.5 \)[/tex].
- For [tex]\( F(3) \)[/tex], use [tex]\( F(3) = F(2) + \frac{1}{2} = 2.5 + \frac{1}{2} = 3.0 \)[/tex].
3. Notice the pattern: Each time we move to the next term, we add [tex]\(\frac{1}{2}\)[/tex].
4. Calculate [tex]\( F(101) \)[/tex]:
- Since we start at 2 for [tex]\( F(1) \)[/tex], and from [tex]\( F(2) \)[/tex] onward we add [tex]\(\frac{1}{2}\)[/tex] each step, the amount added to reach [tex]\( F(101) \)[/tex] will be [tex]\( \frac{1}{2} \times (101-1) = \frac{1}{2} \times 100 = 50 \)[/tex].
5. Find the final value:
- So, [tex]\( F(101) = F(1) + 50 = 2 + 50 = 52 \)[/tex].
The correct answer is [tex]\( C. 52 \)[/tex].
