X ^ 0 + x ^ 1 + x ^ 2 +... x ^ n mod P; X, N, P <= 10 ^ 9

Method 1: quick power. But it must have timed out.

Method 2: Use the proportional sequence formula, but there is division.

Method 3: Sub-governance taste ..

If n is an even number, x ^ 0 + x ^ 1 + x ^ 2 + ....... x ^ n = (x ^ 0 + x ^ 1 + x ^ 2 + ..... x ^ (n/2) * (1 + x ^ (n/2)-x ^ (n/2); that is, F (N) = f (n/2) * (1 + x ^ (n/2)-x ^ (n/2 );

When N is an odd number, x ^ 0 + x ^ 1 + x ^ 2 + ....... x ^ n = (x ^ 0 + x ^ 1 + x ^ 2 + ..... x ^ (n/2) * (1 + x ^ (n/2)-x ^ (n/2) + x ^ N; that is, F (N) = f (n/2) * (1 + x ^ (n/2)-x ^ (n/2) + x ^ N;

The scale of N can be halved in each recursion ..

It's just your own ideas. You are welcome to point out errors or propose better methods.

After the test, the subtraction is also consistent, that is, a % P-B % P + p) % P = (a-B) % P.