Problem:
Score : 400 points

Problem Statement

We have N+1 integers: 10^{100}, 10^{100}+1, ..., 10^{100}+N.

We will choose K or more of these integers. Find the number of possible values of the sum of the chosen numbers, modulo (10^9+7).

Constraints

1 \leq N \leq 2\times 10^5

1 \leq K \leq N+1

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K

Output

Print the number of possible values of the sum, modulo (10^9+7).

Sample Input 1

3 2

Sample Output 1

10

The sum can take 10 values, as follows:

(10^{100})+(10^{100}+1)=2\times 10^{100}+1

(10^{100})+(10^{100}+2)=2\times 10^{100}+2

(10^{100})+(10^{100}+3)=(10^{100}+1)+(10^{100}+2)=2\times 10^{100}+3

(10^{100}+1)+(10^{100}+3)=2\times 10^{100}+4

(10^{100}+2)+(10^{100}+3)=2\times 10^{100}+5

(10^{100})+(10^{100}+1)+(10^{100}+2)=3\times 10^{100}+3

(10^{100})+(10^{100}+1)+(10^{100}+3)=3\times 10^{100}+4

(10^{100})+(10^{100}+2)+(10^{100}+3)=3\times 10^{100}+5

(10^{100}+1)+(10^{100}+2)+(10^{100}+3)=3\times 10^{100}+6

(10^{100})+(10^{100}+1)+(10^{100}+2)+(10^{100}+3)=4\times 10^{100}+6

Sample Input 2

200000 200001

Sample Output 2

1

We must choose all of the integers, so the sum can take just 1 value.

Sample Input 3

141421 35623

Sample Output 3

220280457

Solution in C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using System.Diagnostics;
using System.Text;
using System.Text.RegularExpressions;
using System.Threading.Tasks;
using System.Numerics;
using System.IO;
using System.Runtime.InteropServices;
using static System.Math;
using static Math2;
using static io;
using static Utils;

public class CompetitiveProgramming
{
  public void Solve()

  {
    var (N, K) = Cin<int, int>();
    var fct = new Factorical(N + 1);
    long ans = 0;
    var min = new long[N + 2];
    var max = new long[N + 2];
    for (int i = 1; i < N + 1; ++i)
    {
      min[i] = min[i - 1] + i - 1;
    }
    max[1] = N;
    for (int i = 2; i < N + 1; ++i)
    {
      max[i] = max[i - 1] + N - i + 1;
    }
    for (int i = K; i <= N + 1; ++i)
    {

      ans += max[i] - min[i] + 1;
      ans %= MOD;
    }
    Write(ans);
  }
}

public class Factorical
{
  public long[] Fact;
  public long[] invFact;
  public Factorical(long N, long MOD = 1000000007)
  {
    Fact = new long[N + 1];
    invFact = new long[N + 1];
    Fact[0] = 1;
    for (long i = 1; i <= N; ++i)
    {
      Fact[i] = Fact[i - 1] * i % MOD;
    }
    invFact[N] = Pow(Fact[N], MOD - 2);
    for (long i = N; i > 0; --i)
    {
      invFact[i - 1] = invFact[i] * i % MOD;
    }
  }
  public long Comb(long n, long r, long MOD = 1000000007) => Fact[n] * invFact[r] % MOD * invFact[n - r] % MOD;
}


public static class Math2
{
  public const int INF = 1 << 29;
  public const long INFL = 1L << 60;
  public const int MOD = 1000000007;
  public const int MOD2 = 998244353;

  public static long Pow(long i, long N, long MOD = 1000000007)
  {
    long res = 1;
    while (N > 0)
    {
      if ((N & 1) != 0) res = res * i % MOD;
      i = i * i % MOD;
      N >>= 1;
    }
    return res;
  }
  public static long GCD(long i, long N)
  {
    if (i > N) Swap(ref i, ref N);
    if (i == 0) return N;
    while (N != 0)
    {
      var r = i % N;
      i = N;
      N = r;
    }
    return i;
  }
  public static long LCM(long i, long N) => i * N / GCD(i, N);
  public static long Comb(long N, long R, int MOD = 1000000007)
  {
    long ret = 1;
    long x = 1;
    for (long i = N; i >= N - R + 1; --i)
    {
      ret = ret / x * i;
      ret %= MOD;
      x++;
    }
    return ret;
  }
  public static long Comb2(long N, long R)
  {
    long Nume = 1;
    long Deno = 1;

    if (R > N - R) Swap(ref N, ref R);
    for (long i = 1; i <= R; ++i)
    {
      Deno *= i;
      Nume *= N - i + 1;
    }
    return Deno / Nume;
  }

}
public static class Utils
{
  public static void Swap<T>(ref T A, ref T B)
  {
    T x = A;
    A = B;
    B = x;
  }
  public static int DigitSum(int N)
  {
    string s = N.ToString();
    int ret = 0;
    for (int i = 0; i < s.Length; ++i) ret += s[i] - '0';
    return ret;
  }
}
class Program
{
  static void Main(string[] args)
  {
    var CompetitiveProgramming = new CompetitiveProgramming();
    CompetitiveProgramming.Solve();
  }
}
public static class io
{
  public static string Str => Console.ReadLine();
  public static string[] Strs => Str.Split(' ');
  public static long[] Longs => Strs.Select(long.Parse).ToArray();
  public static int[] Ints => Strs.Select(int.Parse).ToArray();
  public static char[] Chars => Str.ToArray();
  public static double[] Doubles => Strs.Select(double.Parse).ToArray();
  public static long Long1 => Longs[0];
  public static int Int1 => Ints[0];
  public static char Char1 => Chars[0];
  public static double Double1 => Doubles[0];
  public static long[] VerticalRead(int N) { long[] A = new long[N]; for (int i = 0; i < N; ++i) A[i] = Long1; return A; }
  public static void Write(string a) => Console.WriteLine(a);
  public static void Write(params object[] i) => Write(string.Join(" ", i));
  public static void Write<T>(IEnumerable<T> a) => Write(string.Join(" ", a));
  public static void Verticalwrite<T>(IEnumerable<T> a) { foreach (var z in a) Write(z); }
  public static void YN(bool i) { if (i) Write("Yes"); else Write("No"); }


  public static bool IsTypeEqual<T, U>() => typeof(T).Equals(typeof(U));
  public static T ConvertType<T, U>(U a) => (T)Convert.ChangeType(a, typeof(T));
  public static T Cast<T>(string s)
  {
    if (IsTypeEqual<T, int>()) return ConvertType<T, int>(int.Parse(s));
    else if (IsTypeEqual<T, long>()) return ConvertType<T, long>(long.Parse(s));
    else if (IsTypeEqual<T, double>()) return ConvertType<T, double>(double.Parse(s));
    else if (IsTypeEqual<T, char>()) return ConvertType<T, char>(char.Parse(s));
    else return ConvertType<T, string>(s);
  }
  public static (T, U) Cin<T, U>() { var t = Strs; return (Cast<T>(t[0]), Cast<U>(t[1])); }
  public static (T, U, V) Cin<T, U, V>() { var t = Strs; return (Cast<T>(t[0]), Cast<U>(t[1]), Cast<V>(t[2])); }
  public static (T, U, V, W) Cin<T, U, V, W>() { var t = Strs; return (Cast<T>(t[0]), Cast<U>(t[1]), Cast<V>(t[2]), Cast<W>(t[3])); }
  public static (T, U, V, W, X) Cin<T, U, V, W, X>() { var t = Strs; return (Cast<T>(t[0]), Cast<U>(t[1]), Cast<V>(t[2]), Cast<W>(t[3]), Cast<X>(t[4])); }
  public static (T, U, V, W, X, Y) Cin<T, U, V, W, X, Y>() { var t = Strs; return (Cast<T>(t[0]), Cast<U>(t[1]), Cast<V>(t[2]), Cast<W>(t[3]), Cast<X>(t[4]), Cast<Y>(t[5])); }
  public static (T, U, V, W, X, Y, Z) Cin<T, U, V, W, X, Y, Z>() { var t = Strs; return (Cast<T>(t[0]), Cast<U>(t[1]), Cast<V>(t[2]), Cast<W>(t[3]), Cast<X>(t[4]), Cast<Y>(t[5]), Cast<Z>(t[6])); }
}