text: World's most popular travel blog for travel bloggers.
  
  # Finding a solution to the following conditions?
  
  , ,
  Problem Detail:
  
  So for these 5 conditions, I am trying to find the solution/formula for them. What would $a_n$ equal basically? If it helps, the recurrence relation these 5 conditions were generated from was $a_n = a_{n - 1} + 2n$. Any help would be greatly appreciated.
  
  \begin{align*} a_0 &= 4 \\ a_1 &= 6 \\ a_2 &= 10 \\ a_3 &= 16 \\ a_4 &= 24 \end{align*}
  
  ###### Answered By : Yuval Filmus
  
  There are infinitely many sequences starting $4,6,10,16,24$. If I understand you correctly, this sequence was generating according to the rule $a_n = a_{n-1} + 2n$ with the initial value $a_0 = 4$. In that case, you have $$a_n = a_0 + \sum_{k=1}^n (2k) = a_0 + 2 \frac{n(n+1)}{2} = n^2 + n + 4.$$
explanation: The document is a math-related post from what appears to be a Q&A platform. It contains a question about finding a formula for a sequence defined by a recurrence relation, and it also includes an answer that provides the solution to the problem.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: World's most popular travel blog for travel bloggers.
  
  # Finding a solution to the following conditions?
  
  , ,
  Problem Detail:
  
  So for these 5 conditions, I am trying to find the solution/formula for them. What would $a_n$ equal basically? If it helps, the recurrence relation these 5 conditions were generated from was $a_n = a_{n - 1} + 2n$. Any help would be greatly appreciated.
  
  \begin{align*} a_0 &= 4 \\ a_1 &= 6 \\ a_2 &= 10 \\ a_3 &= 16 \\ a_4 &= 24 \end{align*}
  
  ###### Answered By : Yuval Filmus
  
  There are infinitely many sequences starting $4,6,10,16,24$. If I understand you correctly, this sequence was generating according to the rule $a_n = a_{n-1} + 2n$ with the initial value $a_0 = 4$. In that case, you have $$a_n = a_0 + \sum_{k=1}^n (2k) = a_0 + 2 \frac{n(n+1)}{2} = n^2 + n + 4.$$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a math-related post from what appears to be a Q&A platform. It contains a question about finding a formula for a sequence defined by a recurrence relation, and it also includes an answer that provides the solution to the problem.
  
  Label: question-and-answer