text: # The coefficients of 5th, 6th and 7th terms in the expansion
  
  Question:
  
  The coefficients of 5th, 6th and 7th terms in the expansion of (1 + x)n are in A.P., find n.
  
  Solution:
  
  Coefficients of the 5 th, 6 th and 7 th terms in the given expansion are ${ }^{n} C_{4},{ }^{n} C_{5}$ and ${ }^{n} C_{6}$ These coefficients are in $A P$.
  
  Thus, we have
  
  $2^{n} C_{5}={ }^{n} C_{4}+{ }^{n} C_{6}$
  
  On dividing both sides by ${ }^{n} C_{5}$, we get:
  
  $2=\frac{{ }^{n} C_{4}}{{ }^{n} C_{5}}+\frac{{ }^{n} C_{6}}{{ }^{n} C_{5}}$
  
  $\Rightarrow 2=\frac{5}{n-4}+\frac{n-5}{6}$
  
  $\Rightarrow 12 n-48=30+n^{2}-4 n-5 n+20$
  
  $\Rightarrow n^{2}-21 n+98=0$
  
  $\Rightarrow(n-14)(n-7)=0$
  
  $\Rightarrow n=7$ or 14
explanation: The document contains a math question and its corresponding solution. The question asks to find the value of \( n \) for which the coefficients of the 5th, 6th, and 7th terms in the expansion of \( (1 + x)^n \) are in arithmetic progression (A.P.). The solution is provided, leading to the values \( n = 7 \) or \( n = 14 \).
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: # The coefficients of 5th, 6th and 7th terms in the expansion
  
  Question:
  
  The coefficients of 5th, 6th and 7th terms in the expansion of (1 + x)n are in A.P., find n.
  
  Solution:
  
  Coefficients of the 5 th, 6 th and 7 th terms in the given expansion are ${ }^{n} C_{4},{ }^{n} C_{5}$ and ${ }^{n} C_{6}$ These coefficients are in $A P$.
  
  Thus, we have
  
  $2^{n} C_{5}={ }^{n} C_{4}+{ }^{n} C_{6}$
  
  On dividing both sides by ${ }^{n} C_{5}$, we get:
  
  $2=\frac{{ }^{n} C_{4}}{{ }^{n} C_{5}}+\frac{{ }^{n} C_{6}}{{ }^{n} C_{5}}$
  
  $\Rightarrow 2=\frac{5}{n-4}+\frac{n-5}{6}$
  
  $\Rightarrow 12 n-48=30+n^{2}-4 n-5 n+20$
  
  $\Rightarrow n^{2}-21 n+98=0$
  
  $\Rightarrow(n-14)(n-7)=0$
  
  $\Rightarrow n=7$ or 14
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and its corresponding solution. The question asks to find the value of \( n \) for which the coefficients of the 5th, 6th, and 7th terms in the expansion of \( (1 + x)^n \) are in arithmetic progression (A.P.). The solution is provided, leading to the values \( n = 7 \) or \( n = 14 \).
  
  Label: question-and-answer