1. 点查询，区间更新

基础版本

2. 区间更新，点查询

使用差分，维护差分数组d[i] = a[i] - a[i - 1]。
区间更新变成了[l, r] 两端l和r的更新，点查询也就变成了[1, x]的区间更新。

3. 区间更新，区间查询

使用差分，维护差分数组d1[i] = a[i] - a[i - 1] 和 d2[i] = i * (d2[i] - d2[i - 1])。
区间更新的方式和2相同，区间查询是(r + 1) * query(d1, r) - query(d2, r)。通过差分推一推就能得到。

#include <bits/stdc++.h>

using namespace std;


const int maxn = 1e6+10;
ll a[maxn], b[maxn];

ll lowbit(ll t) {  //  core
    return t & (-t);
}

ll n,q;

void update(ll a[], ll x, ll v) {
    for (ll i = x; i <= n; i += lowbit(i)) {
        a[i] += v;
    }
}

ll query(ll a[], ll x) {
    ll res = 0;
    for (ll i = x; i >= 1; i -= lowbit(i)) {
        res += a[i];
    }
    return res;
}

ll query_inner(ll x) {
    return (x + 1) * query(a, x) - query(b, x);
}

ll range_query(ll l, ll r) {
    return query_inner(r) - query_inner(l - 1);
}

int main() {
    scanf("%d%d", &n, &q);
    ll pre = 0;
    for (int i = 1; i <= n; ++i) {
        ll tmp;
        scanf("%lld", &tmp);
        ll d = tmp - pre;
        update(a, i, d);
        update(b, i, d * i);
        pre = tmp;
    }
    for (int i = 0; i < q; ++i) {
        int op;
        scanf("%d", &op);
        if (op == 1) {
            ll l, r;
            ll x;
            scanf("%lld%lld%lld", &l, &r, &x);
            update(a, l, x);
            update(a, r + 1, -x);
            update(b, l, x * l);
            update(b, r + 1, -x * (r + 1));
        }
        else {
            ll l, r;
            scanf("%lld%lld", &l, &r);
            printf("%lld\n", range_query(l, r));
        }
    }

    return 0;
}

4. 二维数状数组

class NumMatrix {
public:
    vector<vector<int>> A;
    int n;
    int m;
    NumMatrix(vector<vector<int>> matrix) {
        n = matrix.size();
        m = matrix[0].size();
        A = vector<vector<int>>(n + 1, vector<int> (m + 1, 0));
        for (int i = 0; i < n; ++i) { 
            for (int j = 0; j < m; ++j) {
                update(i + 1, j + 1, matrix[i][j]);
            }
        }
    }

    void update(int row, int col, int val) {
        for (int i = row; i <= n; i += i&-i) {
            for (int j = col; j <= m; j += j&-j) {
                A[i][j] += val;
            }
        }
    }

    int query(int row, int col) {
        int result = 0;
        for (int i = row; i >= 1; i -= i&-i) {
            for (int j = col; j >= 1; j -= j&-j) {
                result += A[i][j];
            }
        }
        return result;
    }

    int sumRegion(int row1, int col1, int row2, int col2) {
        row1 += 1, col1 += 1, row2 += 1, col2 += 1;
        return query(row2, col2) + query(row1 - 1, col1 - 1) - query(row1 - 1, col2) - query(row2, col1 - 1);
    }
};