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.Numerics;
using System.Collections;
using System.Collections.Generic;
using System.Runtime.CompilerServices;
using System.IO;
using System.Linq;
using System.Text;
using static System.Console;
using static System.Math;
using static Ex;


class AC
{
    /*------------------------------------------------------------------*/

    void Solve()
    {
        io.I(out int N, out int K);
        var cusum = new long[N + 1 + 1];
        var mod = (long)1e9 + 7;
        cusum[1] = 0;
        FOR(2, cusum.Length, i => cusum[i] = (i - 1) + cusum[i - 1]);
        var ans = 0L;
        for (var i = K; i <= N + 1; i++)
        {
            var min = cusum[i];
            var max = cusum[N + 1] - cusum[N + 1 - i];
            var num = max - min + 1;
            ans = (ans + num) % mod;
        }
        WriteLine(ans);
    }
    /*------------------------------------------------------------------*/
    void Test()
    {
        //--- TEST ---//
    }
    public static void Main(string[] args) => new AC().Stream();
    IO io = new IO();
    private void Stream()
    {
        /*
        var exStdIn = new System.IO.StreamReader("stdin.txt");
        SetIn(exStdIn);
        Test();
        */
        Solve();
        io.Flush();
    }
}
class IO
{
    #region INPUT
    #region FastRead
    private const int Size = 1 << 12;
    private readonly Stream _stream = OpenStandardInput();
    private readonly byte[] _buffer = new byte[Size];
    private int _ptr = 0, _buffLen = 0;
    private bool HasNextByte()
    {
        if (_ptr < _buffLen) return true;
        _ptr = 0;
        _buffLen = _stream.Read(_buffer, 0, Size);
        return _buffLen > 0;
    }
    private byte ReadByte() => HasNextByte() ? _buffer[_ptr++] : throw new EndOfStreamException();

    private bool IsAscii(int c) => '!' <= c && c <= '~';
    private char NextChar()
    {
        var b = ReadByte();
        while (!IsAscii(b)) b = ReadByte();
        return (char)b;
    }
    private string Next()
    {
        var sb = new StringBuilder();
        for (var b = NextChar(); IsAscii(b); b = (char)ReadByte()) sb.Append(b);
        return sb.ToString();
    }
    public string String => Next();
    public char Char => NextChar();
    public int Int => int.Parse(Next());
    public long Long => long.Parse(Next());
    public double Double => double.Parse(Next());
    #endregion
    #region LazyRead
    private void Scan<T>(ref T x)
    {
        if (x is int) __refvalue(__makeref(x), int) = int.Parse(Next());
        else if (x is long) __refvalue(__makeref(x), long) = long.Parse(Next());
        else if (x is double) __refvalue(__makeref(x), double) = double.Parse(Next());
        else if (x is char) __refvalue(__makeref(x), char) = NextChar();
        else if (x is uint) __refvalue(__makeref(x), uint) = uint.Parse(Next());
        else if (x is ulong) __refvalue(__makeref(x), ulong) = ulong.Parse(Next());
        else if (x is decimal) __refvalue(__makeref(x), decimal) = decimal.Parse(Next());
        else if (x is BigInteger) __refvalue(__makeref(x), BigInteger) = BigInteger.Parse(Next());
        else __refvalue(__makeref(x), string) = Next();
    }
    public void I<T>(out T t)
    {
        t = default;
        Scan(ref t);
    }

    public void I<T, TU>(out T t, out TU u)
    {
        I(out t);
        I(out u);
    }
    public void I<T, TU, TV>(out T t, out TU u, out TV v)
    {
        I(out t, out u);
        I(out v);
    }
    public void I<T, TU, TV, TW>(out T t, out TU u, out TV v, out TW w)
    {
        I(out t, out u);
        I(out v, out w);
    }
    public void I<T, TU, TV, TW, TX>(out T t, out TU u, out TV v, out TW w, out TX x)
    {
        I(out t, out u, out v);
        I(out w, out x);
    }
    public void I<T, TU, TV, TW, TX, TY>(out T t, out TU u, out TV v, out TW w, out TX x, out TY y)
    {
        I(out t, out u, out v);
        I(out w, out x, out y);
    }
    public void I<T>(out T[] a, int n)
    {
        T[] ts = new T[n];
        a = ts;
        for (int i = 0; i < n; ++i) I(out a[i]);
    }
    public void I<T, TU>(out T[] a, out TU[] b, int n)
    {
        a = new T[n];
        b = new TU[n];
        for (int i = 0; i < n; ++i) I(out a[i], out b[i]);
    }
    public void I<T, TU, TV>(out T[] a, out TU[] b, out TV[] c, int n)
    {
        a = new T[n];
        b = new TU[n];
        c = new TV[n];
        for (var i = 0; i < n; ++i) I(out a[i], out b[i], out c[i]);
    }

    public void I<T, TU, TV, TW>(out T[] a, out TU[] b, out TV[] c, out TW[] d, int n)
    {
        a = new T[n];
        b = new TU[n];
        c = new TV[n];
        d = new TW[n];
        for (var i = 0; i < n; ++i) I(out a[i], out b[i], out c[i], out d[i]);
    }
    public void I<T>(out T[,] a, int h, int w)
    {
        a = new T[h, w];
        for (int i = 0; i < h; ++i) for (int j = 0; j < w; ++j) I(out a[i, j]);
    }
    #endregion
    #endregion
    #region OUTPUT
    private readonly StreamWriter _sw = new StreamWriter(OpenStandardOutput())
    {
#if DEBUG
        AutoFlush = true
#else
        AutoFlush = false
#endif
    };
    public void Flush() => Out.Flush();
    public void YN(bool f) => yn(f, "YES", "NO");
    public void Yn(bool f) => yn(f, "Yes", "No");
    public void yn(bool f, string y = "yes", string n = "no") => WriteLine(f ? y : n);
    #endregion
    public IO() { SetOut(_sw); }
}
static class Ex
{
    #region マクロもどき
    public static T ID<T>(T x) => x;
    public static Dictionary<T1, T2> MERGE<T1, T2>(Dictionary<T1, T2> xs, Dictionary<T1, T2> ys) => xs.Except(xs.Join(ys, z => z.Key, z => z.Key, (a, b) => a)).Concat(ys).ToDictionary(z => z.Key, z => z.Value);
    public static void FOR(int a, int b, Action<int> f) => FOR(a, b, 1, f);
    public static void FOR(int a, int b, int n, Action<int> f)
    {
        for (int i = a; i < b; i += n) f(i);
    }
    public static void REP(int a, Action<int> f) => FOR(0, a, f);
    public static void REPEQ(int a, Action<int> f) => FOR(0, a + 1, f);
    public static void FOREQ(int a, int b, Action<int> f) => FOR(a, b + 1, f);
    public static void FOREQ(int a, int b, int n, Action<int> f) => FOR(a, b + 1, n, f);
    public static void REP(int a, int b, Action<int, int> f) => REP(a, i => REP(b, j => f(i, j)));
    public static void REP(int a, int b, int c, Action<int, int, int> f) => REP(a, i => REP(b, j => REP(c, k => f(i, j, k))));
    public static void REPEQ(int a, int b, Action<int, int> f) => REPEQ(a, i => REPEQ(b, j => f(i, j)));
    public static void REPEQ(int a, int b, int c, Action<int, int, int> f) => REPEQ(a, i => REPEQ(b, j => REPEQ(c, k => f(i, j, k))));
    public static T MIN<T>(params T[] xs) => xs.Min();
    public static T MAX<T>(params T[] xs) => xs.Max();
    public static IEnumerable<int> RANGE(int s, int e) => Enumerable.Range(s, e - s + 1);
    public static int LCM(int a, int b) => a * (b / GCD(a, b));
    public static int GCD(int a, int b) => (a < b) ? GCD(b, a) : b == 0 ? a : GCD(b, a % b);
    public static int LCM(params int[] xs) => xs.Aggregate(LCM);
    public static int GCD(params int[] xs) => xs.Aggregate(GCD);
    public static long LCM(long a, long b) => a * (b / GCD(a, b));
    public static long GCD(long a, long b) => (a < b) ? GCD(b, a) : b == 0L ? a : GCD(b, a % b);
    public static long LCM(params long[] xs) => xs.Aggregate(LCM);
    public static long GCD(params long[] xs) => xs.Aggregate(GCD);
    #endregion
    #region 拡張メソッド
    public static void Rep(this int x, Action<int> f) => REP(x, f);
    public static void RepEq(this int x, Action<int> f) => REPEQ(x, f);
    public static int ToInt(this string s)
    {
        int.TryParse(s.TrimStart('0'), out int n);
        return n;
    }
    public static int ToInt(this char c) => ToInt(c.ToString());
    public static int ToInt(this char[] c) => ToInt(new string(c));
    public static long ToLong(this string s)
    {
        long.TryParse(s.TrimStart('0'), out long n);
        return n;
    }
    public static long ToLong(this int x) => (long)x;
    public static IEnumerable<long> ToLong(this IEnumerable<int> xs) => xs.Select(x => x.ToLong());
    public static long ToLong(this char c) => ToLong(c.ToString());
    public static long ToLong(this char[] c) => ToLong(new string(c));
    public static int ToMod(this int x, int mod = (int)(1e9 + 7)) => x % mod;
    public static long ToMod(this long x, long mod = (long)(1e9 + 7)) => x % mod;
    public static void Do<T>(this T x, Action<T> f) => f(x);
    public static TU Then<T, TU>(this T x, Func<T, TU> f) => f(x);
    public static bool IsIn<T>(this T[] xs, int i) => 0 <= i && i < xs.Length;
    public static bool IsIn<T>(this T[,] xxs, int i, int j) => 0 <= i && i < xxs.GetLength(0) && 0 <= j && j < xxs.GetLength(1);
    public static void Iter<T>(this IEnumerable<T> xs, Action<T> f)
    {
        foreach (var x in xs) f(x);
    }
    public struct IndexedItem<T>
    {
        public int Index;
        public T Value;
        public IndexedItem(int idx, T val) => (Index, Value) = (idx, val);
    }
    public static IEnumerable<IndexedItem<T>> Indexed<T>(this IEnumerable<T> xs) => xs.Select((x, i) => new IndexedItem<T>(i, x));
    public static void Set<T>(this T[] xs, T v) => REP(xs.Length, i => xs[i] = v);
    public static void Set<T>(this T[,] xxs, T v) => REP(xxs.GetLength(0), i => REP(xxs.GetLength(1), j => xxs[i, j] = v));
    public static string Pad0<T>(this T s, int n) => s.ToString().PadLeft(n, '0');
    public static int RoundOff(this double v, int n) => (int)Round(v, n - 1, MidpointRounding.AwayFromZero);
    public static int RoundUp(this double v) => (int)Ceiling(v);
    public static int RoundDown(this double v) => (int)Floor(v);
    public static bool Odd(this int v) => (v & 1) != 0;
    public static bool Odd(this long v) => (v & 1) != 0;
    public static bool Eve(this int v) => !v.Odd();
    public static bool Eve(this long v) => !v.Odd();
    public static void Prt<T>(this T x) => WriteLine(x);
    public static void Prt<T>(this T[] xs) => xs.Iter(x => WriteLine(x));
    public static void Prt<T>(this List<T> xs) => xs.Iter(x => WriteLine(x));
    public static void Prt<T>(this T[,] xs) => xs.GetLength(0).Rep(i =>
    {
        xs.GetLength(1).Rep(j => Write($"{xs[i, j]} "));
        WriteLine();
    });
    public static void Prt_<T>(this T[] xs) => WriteLine(string.Join(" ", xs));
    public static void Log<T>(this T x, [CallerLineNumber] int line = 0)
    {
#if DEBUG
        Error.WriteLine($"--- LOG[{line}]: {x} ---");
#endif
    }
    public static void Log<T>(this T[] xs, [CallerLineNumber] int line = 0)
    {
#if DEBUG
        Error.WriteLine($"^^^ LOG[{line}] ^^^");
        xs.Iter(x => Error.WriteLine(x));
        Error.WriteLine($"vvv LOG[{line}] vvv\n");
#endif
    }
    public static void Log<T>(this T[,] xs, [CallerLineNumber] int line = 0)
    {
#if DEBUG
        Error.WriteLine($"^^^ LOG[{line}] ^^^");
        xs.GetLength(0).Rep(i =>
        {
            xs.GetLength(1).Rep(j => Error.Write($"{xs[i, j]} "));
            Error.WriteLine();
        });
        Error.WriteLine($"vvv LOG[{line}] vvv\n");
#endif
    }
    public static string Reversed(this string s) => string.Join("", s.Reverse());
    public static string Repeat(this string s, int n) => string.Concat(Enumerable.Repeat(s, n));
    #endregion
}