// Example: left = 2, right = 8
// Explanation: F(2) + F(3) + F(4) + F(5) + F(6) + F(7) + F(8) = 1 + 2 + 3 + 5 + 8 + 13 + 21 = 53
// Function to compute Fibonacci numbers up to n
function fibonacci(int $n): int {
if ($n === 0) return 0;
if ($n === 1) return 1;
$prev = 0;
$curr = 1;
for ($i = 2; $i <= $n; $i++) {
$next = $prev + $curr;
$prev = $curr;
$curr = $next;
}
return $curr;
}
// Function to calculate sum of Fibonacci numbers from index L to R (inclusive)
function sumFibonacciRange(int $L, int $R): int {
// If the range is invalid (e.g., L > R), return 0
if ($L > $R) return 0;
/*
Mathematical identity:
Sum(F_L + F_(L+1) + ... + F_R) = F_(R+2) - F_(L+1)
Explanation:
- The sum of the first n Fibonacci numbers is F_(n+2) - 1.
- So, the sum from F_L to F_R can be derived by subtracting:
(sum of first R terms) - (sum of first (L-1) terms)
= (F_(R+2) - 1) - (F_(L+1) - 1)
= F_(R+2) - F_(L+1)
*/
return fibonacci($R + 2) - fibonacci($L + 1);
}
// Main program
$left = 2;
$right = 8;
$result = sumFibonacciRange($left, $right);
echo "Sum of Fibonacci numbers from index $left to $right = $result\n";
/*
run:
Sum of Fibonacci numbers from index 2 to 8 = 53
*/
