1852A · Ntarsis' Set

Problem: Ntarsis has been given a set $S$, initially containing integers $1, 2, 3, \ldots, 10^{1000}$ in sorted order. Every day, he will remove the $a_1$-th, $a_2$-th, $\ldots$, $a_n$-th smallest numbers in $S$ simultaneously.

What is the smallest element in $S$ after $k$ days?

Input Format: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^5$). The description of the test cases follows.

The first line of each test case consists of two integers $n$ and $k$ ($1 \leq n,k \leq 2 \cdot 10^5$) — the length of $a$ and the number of days.

The following line of each test case consists of $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \leq a_i \leq 10^9$) — the elements of array $a$.

It is guaranteed that:

• The sum of $n$ over all test cases won't exceed $2 \cdot 10^5$;
• The sum of $k$ over all test cases won't exceed $2 \cdot 10^5$;
• $a_1 < a_2 < \cdots < a_n$ for all test cases.

Output Format: For each test case, print an integer that is the smallest element in $S$ after $k$ days.

Note: For the first test case, each day the $1$-st, $2$-nd, $4$-th, $5$-th, and $6$-th smallest elements need to be removed from $S$. So after the first day, $S$ will become $\require{cancel}$ $\{\cancel 1, \cancel 2, 3, \cancel 4, \cancel 5, \cancel 6, 7, 8, 9, \ldots\} = \{3, 7, 8, 9, \ldots\}$. The smallest element is $3$.

For the second case, each day the $1$-st, $3$-rd, $5$-th, $6$-th and $7$-th smallest elements need to be removed from $S$. $S$ will be changed as follows:

Day$S$ before$S$ after1\{\cancel 1, 2, \cancel 3, 4, \cancel 5, \cancel 6, \cancel 7, 8, 9, 10, \ldots \}$$$$$$\to$$$$$$\{2, 4, 8, 9, 10, \ldots\}2\{\cancel 2, 4, \cancel 8, 9, \cancel{10}, \cancel{11}, \cancel{12}, 13, 14, 15, \ldots\}$$$$$$\to$$$$$$\{4, 9, 13, 14, 15, \ldots\}3\{\cancel 4, 9, \cancel{13}, 14, \cancel{15}, \cancel{16}, \cancel{17}, 18, 19, 20, \ldots\}$$$$$$\to$$$$$$\{9, 14, 18, 19, 20, \ldots\}

The smallest element left after $k = 3$ days is $9$.