Given two numbers and . The task is to find the sum of the sequence given below.
(1*2*3*…*k) + (2*3*…*k*(k+1)) + (3*4*..*(k+1)*(k+2)) +…..+((n-k+1)*(n-k+2)*…*(n-k+k)).
Since the output can be large, print the answer under modulo 10^9+7.
Input : N = 3, K = 2 Output : 8 Input : N = 4, K = 2 Output : 20
Let us take the given example and try to reduce it to a general formula.
In the given example for n = 3 and k=2,
Sum = 1*2 + 2*3
We know that:
So each term is of the form:
If we multiply and divide by , it becomes
Which is nothing but,
But since n is so large we can not calculate it directly, we have to simplify the above expression.
On Simplifying we get,
Below is the implementation of the above idea:
$n – $k; $i–)
$ans = $ans * $i % $MOD;
$ans = $ans * modInv($k + 1) % $MOD;
// Driver code
$n = 3; $k = 2;
echo getSum($n, $k);
// This code is contributed
// by Akanksha Rai
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