{"problem":{"name":"Journey","description":{"content":"We have a graph with $N$ vertices called Vertex $1$ through $N$. Takahashi is standing on Vertex $1$.   The graph has no edges now.   Takahashi will repeatedly do the following operation: 1.  Choose ","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"abc194_d"},"statements":[{"statement_type":"Markdown","content":"We have a graph with $N$ vertices called Vertex $1$ through $N$. Takahashi is standing on Vertex $1$.  \nThe graph has no edges now.  \nTakahashi will repeatedly do the following operation:\n\n1.  Choose one of the $N$ vertices (including the one on which Takahashi is standing now). Every vertex is chosen with probability $\\frac{1}{N}$, independently from previous operations.\n2.  Add an edge between the vertex on which Takahashi is standing now and the chosen vertex, and go to the chosen vertex.\n\nFind the expected value of the number of times he does the operation until the graph becomes connected.\n\n## Constraints\n\n*   $2 \\le N \\le 10^5$\n\n## Input\n\nInput is given from Standard Input in the following format:\n\n$N$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"abc194_d","tags":[],"sample_group":[["2","2.00000000000\n\nThe graph becomes connected when the operation chooses Vertex $2$ for the first time.  \nBy considering the case Vertex $2$ is chosen for the first time in the $i$\\-th operation for each $i$, the answer is $\\sum_{i = 1}^{\\infty} (i \\times (\\frac{1}{2})^i) = 2$."],["3","4.50000000000"]],"created_at":"2026-03-03 11:01:14"}}