Answer :
To find the value of [tex]\( F(101) \)[/tex] given that [tex]\( F(1) = 2 \)[/tex] and [tex]\( F(n) = F(n-1) + \frac{1}{2} \)[/tex] for all integer numbers [tex]\( n \geq 1 \)[/tex], follow these steps:
1. Start with the given:
[tex]\( F(1) = 2 \)[/tex].
2. Understand the recursive formula:
The formula [tex]\( F(n) = F(n-1) + \frac{1}{2} \)[/tex] means that each term in the sequence is obtained by adding [tex]\(\frac{1}{2}\)[/tex] to the previous term.
3. Find a pattern or general formula:
Let's calculate a few terms to spot any pattern:
- [tex]\( F(2) = F(1) + \frac{1}{2} = 2 + \frac{1}{2} = 2.5 \)[/tex]
- [tex]\( F(3) = F(2) + \frac{1}{2} = 2.5 + \frac{1}{2} = 3 \)[/tex]
- [tex]\( F(4) = F(3) + \frac{1}{2} = 3 + \frac{1}{2} = 3.5 \)[/tex]
From these examples, you can see that each additional term increases by [tex]\(\frac{1}{2}\)[/tex].
4. Derive a formula for [tex]\( F(n) \)[/tex]:
Start with [tex]\( F(1) = 2 \)[/tex]. Each step increases the value by [tex]\(\frac{1}{2}\)[/tex]. For [tex]\( F(n) \)[/tex], there are [tex]\( n - 1 \)[/tex] steps from [tex]\( F(1) \)[/tex] to [tex]\( F(n) \)[/tex]. So, the formula becomes:
[tex]\[
F(n) = 2 + (n - 1) \times \frac{1}{2}
\][/tex]
5. Calculate [tex]\( F(101) \)[/tex]:
Substituting [tex]\( n = 101 \)[/tex] into the derived formula:
[tex]\[
F(101) = 2 + (101 - 1) \times \frac{1}{2}
\][/tex]
[tex]\[
= 2 + 100 \times \frac{1}{2}
\][/tex]
[tex]\[
= 2 + 50
\][/tex]
[tex]\[
= 52
\][/tex]
6. Conclusion:
Thus, the value of [tex]\( F(101) \)[/tex] is [tex]\(\boxed{52}\)[/tex].
1. Start with the given:
[tex]\( F(1) = 2 \)[/tex].
2. Understand the recursive formula:
The formula [tex]\( F(n) = F(n-1) + \frac{1}{2} \)[/tex] means that each term in the sequence is obtained by adding [tex]\(\frac{1}{2}\)[/tex] to the previous term.
3. Find a pattern or general formula:
Let's calculate a few terms to spot any pattern:
- [tex]\( F(2) = F(1) + \frac{1}{2} = 2 + \frac{1}{2} = 2.5 \)[/tex]
- [tex]\( F(3) = F(2) + \frac{1}{2} = 2.5 + \frac{1}{2} = 3 \)[/tex]
- [tex]\( F(4) = F(3) + \frac{1}{2} = 3 + \frac{1}{2} = 3.5 \)[/tex]
From these examples, you can see that each additional term increases by [tex]\(\frac{1}{2}\)[/tex].
4. Derive a formula for [tex]\( F(n) \)[/tex]:
Start with [tex]\( F(1) = 2 \)[/tex]. Each step increases the value by [tex]\(\frac{1}{2}\)[/tex]. For [tex]\( F(n) \)[/tex], there are [tex]\( n - 1 \)[/tex] steps from [tex]\( F(1) \)[/tex] to [tex]\( F(n) \)[/tex]. So, the formula becomes:
[tex]\[
F(n) = 2 + (n - 1) \times \frac{1}{2}
\][/tex]
5. Calculate [tex]\( F(101) \)[/tex]:
Substituting [tex]\( n = 101 \)[/tex] into the derived formula:
[tex]\[
F(101) = 2 + (101 - 1) \times \frac{1}{2}
\][/tex]
[tex]\[
= 2 + 100 \times \frac{1}{2}
\][/tex]
[tex]\[
= 2 + 50
\][/tex]
[tex]\[
= 52
\][/tex]
6. Conclusion:
Thus, the value of [tex]\( F(101) \)[/tex] is [tex]\(\boxed{52}\)[/tex].
