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23. If [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], then find the value of [tex]$F(101)$[/tex].

A. 49
B. 50
C. 52
D. 51

Answer :

To solve the problem, we need to find the value of [tex]\( F(101) \)[/tex] given that:

- [tex]\( F(1) = 2 \)[/tex]
- [tex]\( F(n) = F(n-1) + \frac{1}{2} \)[/tex] for all integer numbers [tex]\( n \geq 1 \)[/tex]

The given conditions describe an arithmetic sequence where:

- The first term [tex]\( F(1) = 2 \)[/tex]
- Each subsequent term increases by [tex]\( \frac{1}{2} \)[/tex]

We are tasked with finding [tex]\( F(101) \)[/tex].

Steps to find [tex]\( F(101) \)[/tex]:

1. Identify the Initial Term:
[tex]\[
F(1) = 2
\][/tex]

2. Determine the Common Difference:
[tex]\[
d = \frac{1}{2}
\][/tex]
This is the amount added to each term to get the next term in the sequence.

3. Calculate the Number of Steps from [tex]\( F(1) \)[/tex] to [tex]\( F(101) \)[/tex]:
Since we start at [tex]\( n=1 \)[/tex] and want to find [tex]\( n=101 \)[/tex], the number of steps is:
[tex]\[
101 - 1 = 100
\][/tex]

4. Calculate the Total Increase from [tex]\( F(1) \)[/tex] to [tex]\( F(101) \)[/tex]:
The total increase over 100 steps is:
[tex]\[
\text{Total increase} = 100 \times \frac{1}{2} = 50
\][/tex]

5. Determine [tex]\( F(101) \)[/tex]:
Add the total increase to the initial term:
[tex]\[
F(101) = F(1) + \text{Total increase} = 2 + 50 = 52
\][/tex]

Hence, the value of [tex]\( F(101) \)[/tex] is 52. Therefore, the correct answer is:

C. 52

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