七年级含参不等式整数解最值问题

问题

已知${\displaystyle \frac{7}{13}}$<${\displaystyle \frac{n}{n+k}}$<${\displaystyle \frac{6}{11}}$，其中$n$、$k$为正整数。
（$1$）对于某一个$n$，若只有$1$个$k$使不等式${\displaystyle \frac{7}{13}}$<${\displaystyle \frac{n}{n+k}}$<${\displaystyle \frac{6}{11}}$成立，则$n$的最小值和最大值分别是多少？
（$2$）对于某一个$n$，若只有$3$个$k$使不等式${\displaystyle \frac{7}{13}}$<${\displaystyle \frac{n}{n+k}}$<${\displaystyle \frac{6}{11}}$成立，则$n$的最小值和最大值分别是多少？

解析

先考虑最大

$\because {\displaystyle \frac{7}{13}}<{\displaystyle \frac{n}{n+k}}<{\displaystyle \frac{6}{11}}$

$\therefore {\displaystyle \frac{13}{7}}>{\displaystyle \frac{n+k}{n}}>{\displaystyle \frac{11}{6}}$

$\therefore {\displaystyle \frac{6}{7}}>{\displaystyle \frac{k}{n}}>{\displaystyle \frac{5}{6}}$

$\therefore {\displaystyle \frac{5}{6}}n<k<{\displaystyle \frac{6}{7}}n$

若只有$1$个$k$成立，则 ${\displaystyle \frac{6}{7}}n-{\displaystyle \frac{5}{6}}n⩽2\Rightarrow {\displaystyle \frac{1}{42}}n⩽2\Rightarrow n⩽84$

若只有$3$个$k$成立，则 ${\displaystyle \frac{6}{7}}n-{\displaystyle \frac{5}{6}}n⩽4\Rightarrow {\displaystyle \frac{1}{42}}n⩽4\Rightarrow n⩽168$

再考虑最小

$\because {\displaystyle \frac{6}{7}}>{\displaystyle \frac{k}{n}}>{\displaystyle \frac{5}{6}}$

$\therefore \{\begin{array}{l}6n>7k\\ 6k>5n\end{array}$

$\therefore \{\begin{array}{l}6n-1⩾7k\\ 6k⩾5n+1\end{array}$

$\therefore {\displaystyle \frac{6n-1}{7}}⩾k⩾{\displaystyle \frac{5n+1}{6}}$

若只有$1$个$k$成立，则${\displaystyle \frac{6n-1}{7}}-{\displaystyle \frac{5n+1}{6}}⩾0\Rightarrow n⩾13$

若只有$3$个$k$成立，则${\displaystyle \frac{6n-1}{7}}-{\displaystyle \frac{5n+1}{6}}⩾2\Rightarrow n⩾97$

经验证都成立

代码验证

result = []  # 用于保存符合条件的结果

for n in range(1, 301):  # 枚举 n 从 1 到 300
    lower = 5 * n / 6  # 计算下界
    upper = 6 * n / 7  # 计算上界
    k_values=[]
    count = 0
    for k in range(int(lower) , int(upper)+1):  # 枚举 k 从下界向上取整后再加 1，到上界向下取整后的整数部分之间
        if lower < k < upper:  # 如果 k 符合条件
            count += 1  # 计算出现的次数，并记录到列表中。
            k_values.append(k)  # 将符合条件的结果加入列表
    n_list = [f"n={n}", f"{lower:.2f}<k<{upper:.2f}"]
    k_count = [f"一共{count}个k"]
    n_list.extend(k_values)  # 将 k 值列表加入到 n_list 列表中
    n_list.extend(k_count)  # 将计数列表加入到 n_list 列表中
    result.append(n_list)  # 将 n_list 加入到结果列表中

# 输出结果
print(result)

['n=1', '0.83<k<0.86', '一共0个k'], 
['n=2', '1.67<k<1.71', '一共0个k'], 
['n=3', '2.50<k<2.57', '一共0个k'], 
['n=13', '10.83<k<11.14', 11, '一共1个k'], 
['n=14', '11.67<k<12.00', '一共0个k'], 
['n=84', '70.00<k<72.00', 71, '一共1个k'], 
['n=85', '70.83<k<72.86', 71, 72, '一共2个k'], 
['n=86', '71.67<k<73.71', 72, 73, '一共2个k'], 
['n=87', '72.50<k<74.57', 73, 74, '一共2个k'], 
['n=96', '80.00<k<82.29', 81, 82, '一共2个k'], 
['n=97', '80.83<k<83.14', 81, 82, 83, '一共3个k'], 
['n=98', '81.67<k<84.00', 82, 83, '一共2个k'], 
['n=99', '82.50<k<84.86', 83, 84, '一共2个k'], 
['n=168', '140.00<k<144.00', 141, 142, 143, '一共3个k'], 
['n=169', '140.83<k<144.86', 141, 142, 143, 144, '一共4个k'], 
['n=170', '141.67<k<145.71', 142, 143, 144, 145, '一共4个k'], 
['n=171', '142.50<k<146.57', 143, 144, 145, 146, '一共4个k'], 
['n=172', '143.33<k<147.43', 144, 145, 146, 147, '一共4个k'],