Enter a number to check if it belongs to the Fibonacci sequence, or input \( N \) to calculate the \( N \)th term and the sum of the first \( N \) terms.
The Fibonacci sequence is a series of numbers where each term is the sum of the two preceding ones, starting from 0 and 1:
To check if a number \( x \) is a Fibonacci number, it must satisfy one of the following conditions: \( 5x^2 + 4 \) \( 5x^2 - 4 \) At least one of these results must be a perfect square. If true, \( x \) is a Fibonacci number.
Solution:
Compute \( 5 \times 13^2 + 4 = 5 \times 169 + 4 = 845 + 4 = 849 \).
Compute \( 5 \times 13^2 - 4 = 5 \times 169 - 4 = 845 - 4 = 841 \).
841 is a perfect square（\( \sqrt{841} = 29 \)
Result: 13 is a Fibonacci number.
Solution:
Compute \( 5 \times 101^2 + 4 = 5 \times 10201 + 4 = 51005 + 4 = 51009 \).
Compute \( 5 \times 101^2 - 4 = 5 \times 10201 - 4 = 51005 - 4 = 51001 \).
Neither \( 51009 \) nor \( 51001 \) is a perfect square.
Result: 101 is not a Fibonacci number.