Score : $500$ points

问题陈述

让我们用一个骰子来玩一个游戏。游戏最多包含 $N$ 轮，每轮的规则如下：

• 投掷一个六面骰子，其上的数字 $1,\ldots,6$ 出现的概率相等，并令 $X$ 为所显示的数字（每次投掷与其他投掷相互独立）。
• 如果现在是第 $N$ 轮，你的 得分 为 $X$，游戏结束。
• 否则，你可以选择继续游戏或结束游戏。
  • 如果你选择结束游戏，你的得分即为 $X$，且不再进行更多轮次。

当你以最大化期望值的方式来玩游戏时，求你的得分的期望值。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

Let us play a game using a die. The game consists of at most $N$ turns, each of which goes as follows.

• Throw a $6$-sided die that shows $1,\ldots,6$ with equal probability, and let $X$ be the number shown (each throw is independent of the others).
• If it is the $N$-th turn now, your score is $X$, and the game ends.
• Otherwise, choose whether to continue or end the game.
  • If you end the game, your score is $X$, and there is no more turn.

Find the expected value of your score when you play the game to maximize this expected value.

Constraints

• $1 \leq N \leq 100$

Input

Input is given from Standard Input in the following format:

$N$

Output

Print the answer.
Your output is considered correct if its absolute or relative error from the true answer is at most $10^{-6}$.

Sample Input 1

1

Sample Output 1

3.5000000000

Sample Input 2

2

Sample Output 2

4.2500000000

Sample Input 3

10

Sample Output 3

5.6502176688

update @ 2024/3/10 11:15:06