To find the value of [tex]\( n \)[/tex] for which [tex]\( x_n + x_{n+1} = 55 \)[/tex] in the Fibonacci sequence with the initial values [tex]\( x_1 = 1 \)[/tex], [tex]\( x_2 = 1 \)[/tex], and [tex]\( x_3 = 2 \)[/tex], we need to generate the Fibonacci sequence and check the sums accordingly.
### Step-by-Step Solution
1. Initialize the first three Fibonacci numbers:
- [tex]\( x_1 = 1 \)[/tex]
- [tex]\( x_2 = 1 \)[/tex]
- [tex]\( x_3 = 2 \)[/tex]
2. Generate the Fibonacci sequence beyond the first three terms:
- [tex]\( x_4 = x_3 + x_2 = 2 + 1 = 3 \)[/tex]
- [tex]\( x_5 = x_4 + x_3 = 3 + 2 = 5 \)[/tex]
- [tex]\( x_6 = x_5 + x_4 = 5 + 3 = 8 \)[/tex]
- [tex]\( x_7 = x_6 + x_5 = 8 + 5 = 13 \)[/tex]
- [tex]\( x_8 = x_7 + x_6 = 13 + 8 = 21 \)[/tex]
- [tex]\( x_9 = x_8 + x_7 = 21 + 13 = 34 \)[/tex]
3. Evaluate the sums of consecutive Fibonacci numbers:
- For [tex]\( n = 1 \)[/tex]: [tex]\( x_1 + x_2 = 1 + 1 = 2 \)[/tex]
- For [tex]\( n = 2 \)[/tex]: [tex]\( x_2 + x_3 = 1 + 2 = 3 \)[/tex]
- For [tex]\( n = 3 \)[/tex]: [tex]\( x_3 + x_4 = 2 + 3 = 5 \)[/tex]
- For [tex]\( n = 4 \)[/tex]: [tex]\( x_4 + x_5 = 3 + 5 = 8 \)[/tex]
- For [tex]\( n = 5 \)[/tex]: [tex]\( x_5 + x_6 = 5 + 8 = 13 \)[/tex]
- For [tex]\( n = 6 \)[/tex]: [tex]\( x_6 + x_7 = 8 + 13 = 21 \)[/tex]
- For [tex]\( n = 7 \)[/tex]: [tex]\( x_7 + x_8 = 13 + 21 = 34 \)[/tex]
- For [tex]\( n = 8 \)[/tex]: [tex]\( x_8 + x_9 = 21 + 34 = 55 \)[/tex]
4. Identify the value of [tex]\( n \)[/tex] where the sum equals 55:
- We find that [tex]\( n = 8 \)[/tex] satisfies the condition [tex]\( x_n + x_{n+1} = 55 \)[/tex].
Therefore, the value of [tex]\( n \)[/tex] for which [tex]\( x_n + x_{n+1} = 55 \)[/tex] in the given sequence is:
[tex]\[ n = 8 \][/tex]
This completes our solution.
