To determine which Fibonacci number results from adding [tex]\( F(51) \)[/tex] and [tex]\( F(52) \)[/tex], we need to follow the properties of the Fibonacci sequence.
Here's a clear step-by-step approach:
1. Identify the given Fibonacci numbers:
- [tex]\( F(51) = 20,365,011,074 \)[/tex]
- [tex]\( F(52) = 32,951,280,099 \)[/tex]
2. Add the given Fibonacci numbers:
- [tex]\( F(51) + F(52) \)[/tex]
3. Perform the addition:
[tex]\[
20,365,011,074 + 32,951,280,099 = 53,316,291,173
\][/tex]
4. Identify the result in the context of the Fibonacci sequence:
In the Fibonacci sequence, each number is the sum of the two preceding numbers. Therefore:
[tex]\[
F(n) = F(n-1) + F(n-2)
\][/tex]
Using this rule:
[tex]\[
F(53) = F(52) + F(51)
\][/tex]
5. Compare your result with the step-by-step calculation:
- The result of adding [tex]\( F(51) \)[/tex] and [tex]\( F(52) \)[/tex] is [tex]\( 53,316,291,173 \)[/tex], which fits the position [tex]\( F(53) \)[/tex] in the sequence.
Thus, when [tex]\( F(51) \)[/tex] and [tex]\( F(52) \)[/tex] (i.e., [tex]\( 20,365,011,074 \)[/tex] and [tex]\( 32,951,280,099 \)[/tex]) are added together, the result is the Fibonacci number [tex]\( F(53) \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{F(53)} \][/tex]
