text: ﻿ Seven numbers a_1, a_2, a_3, a_4, a_5, a_6 and a_7 form an arithmetic sequence, where
  
  #### Question
  
  Seven numbers a_1, a_2, a_3, a_4, a_5, a_6 and a_7 form an arithmetic sequence, where a_1 < a_2 < a_3 < a_4 < a_5 < a_6 < a_7. It is given that the mean of the seven numbers is 0.
  
  Find a_4.
  
  If the number a_4 is deleted, is there any change in the variance of the numbers? Explain your answer.
  
  Collected in the board: Arithmetic sequence
  
  Steven Zheng posted 1 year ago
  
  Let d be the common difference of the sequence
  
  a_1=a_4-3d
  
  a_2 = a_4-2d
  
  a_3 = a_4-d
  
  a_5=a_4+d
  
  a_6=a_4+2d
  
  a_7 = a_4+3d
  
  a_1+a_2+\dots+a_7
  
  =7a_4=0
  
  Therefore,
  
  a_4 = 0
  
  If a_4 is deleted, previous a_5 becomes the 4th term, previous a_6 becomes the 5th term, previous a_7 becomes 6th term.
  
  a_1=a_4-3d
  
  a_2 = a_4-2d
  
  a_3 = a_4-d
  
  a_4=a_4
  
  a_5=a_4+d
  
  a_6 = a_4+2d
  
  a_1+a_2+\dots+a_6
  
  =6a_4-3d=0
  
  Therefore,
  
  a_4=\dfrac{1}{2}d
  
  which means the numbers vary with the common difference.
  
  Steven Zheng posted 1 year ago
  
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explanation: The document contains a math question and an answer to that question. The question involves an arithmetic sequence and asks for the value of \( a_4 \) and whether the variance of the numbers changes if \( a_4 \) is deleted. The answer provides a detailed step-by-step solution to both parts of the question.
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: ﻿ Seven numbers a_1, a_2, a_3, a_4, a_5, a_6 and a_7 form an arithmetic sequence, where
  
  #### Question
  
  Seven numbers a_1, a_2, a_3, a_4, a_5, a_6 and a_7 form an arithmetic sequence, where a_1 < a_2 < a_3 < a_4 < a_5 < a_6 < a_7. It is given that the mean of the seven numbers is 0.
  
  Find a_4.
  
  If the number a_4 is deleted, is there any change in the variance of the numbers? Explain your answer.
  
  Collected in the board: Arithmetic sequence
  
  Steven Zheng posted 1 year ago
  
  Let d be the common difference of the sequence
  
  a_1=a_4-3d
  
  a_2 = a_4-2d
  
  a_3 = a_4-d
  
  a_5=a_4+d
  
  a_6=a_4+2d
  
  a_7 = a_4+3d
  
  a_1+a_2+\dots+a_7
  
  =7a_4=0
  
  Therefore,
  
  a_4 = 0
  
  If a_4 is deleted, previous a_5 becomes the 4th term, previous a_6 becomes the 5th term, previous a_7 becomes 6th term.
  
  a_1=a_4-3d
  
  a_2 = a_4-2d
  
  a_3 = a_4-d
  
  a_4=a_4
  
  a_5=a_4+d
  
  a_6 = a_4+2d
  
  a_1+a_2+\dots+a_6
  
  =6a_4-3d=0
  
  Therefore,
  
  a_4=\dfrac{1}{2}d
  
  which means the numbers vary with the common difference.
  
  Steven Zheng posted 1 year ago
  
  Scroll to Top
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and an answer to that question. The question involves an arithmetic sequence and asks for the value of \( a_4 \) and whether the variance of the numbers changes if \( a_4 \) is deleted. The answer provides a detailed step-by-step solution to both parts of the question.
  
  Label: question-and-answer