text: ruigE
  
  2021-08-21
  
  Consider the following sequences
  1)${a}_{n}=-2,5,12,19,\dots$
  2)${b}_{n}=3,6,12,24,48\dots$
  Find the next two terms of the sequence
  Find a recurrence relation that generates the sequence
  Find an explicit formula for the nth term of the sequence
  
  irwchh
  
  1)
  Given sequence in an airthematic sequence
  First term, ${a}_{1}=-2$
  Common difference, $d=5-\left(-2\right)=5+2=7$
  Next two terms are ${a}_{5}={a}_{1}\left(5-1\right)d=-2+\left(4\right)7=-2+28$
  ${a}_{5}=26$
  ${a}_{6}={a}_{1}\left(6-1\right)d=-2+\left(5\right)7=-2+35$
  ${a}_{5}=33$
  Recurrence formula
  ${a}_{1}=-2$
  ${a}_{n}={a}_{n-1}+7$
  Explicit formula
  ${a}_{5}={a}_{1}\left(n-1\right)d$
  2)
  Given sequence is a multiple of 2
  First term ${a}_{1}=3$
  ${a}_{6}=2{a}_{5}=2\left(48\right)=96$
  ${a}_{7}=2{a}_{6}=2\left(96\right)=192$
  Recurrence formula
  ${a}_{1}=3$
  ${a}_{n}=2{a}_{n-1}$
  Explicit formula ${a}_{n}=3{\left(2\right)}^{n-1}$
  
  Do you have a similar question?
  
  Recalculate according to your conditions!
explanation: The document contains a math problem with a detailed solution. It presents two sequences and asks for the next two terms, a recurrence relation, and an explicit formula for each sequence. The solution is provided for both sequences, including the calculations and formulas.
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: ruigE
  
  2021-08-21
  
  Consider the following sequences
  1)${a}_{n}=-2,5,12,19,\dots$
  2)${b}_{n}=3,6,12,24,48\dots$
  Find the next two terms of the sequence
  Find a recurrence relation that generates the sequence
  Find an explicit formula for the nth term of the sequence
  
  irwchh
  
  1)
  Given sequence in an airthematic sequence
  First term, ${a}_{1}=-2$
  Common difference, $d=5-\left(-2\right)=5+2=7$
  Next two terms are ${a}_{5}={a}_{1}\left(5-1\right)d=-2+\left(4\right)7=-2+28$
  ${a}_{5}=26$
  ${a}_{6}={a}_{1}\left(6-1\right)d=-2+\left(5\right)7=-2+35$
  ${a}_{5}=33$
  Recurrence formula
  ${a}_{1}=-2$
  ${a}_{n}={a}_{n-1}+7$
  Explicit formula
  ${a}_{5}={a}_{1}\left(n-1\right)d$
  2)
  Given sequence is a multiple of 2
  First term ${a}_{1}=3$
  ${a}_{6}=2{a}_{5}=2\left(48\right)=96$
  ${a}_{7}=2{a}_{6}=2\left(96\right)=192$
  Recurrence formula
  ${a}_{1}=3$
  ${a}_{n}=2{a}_{n-1}$
  Explicit formula ${a}_{n}=3{\left(2\right)}^{n-1}$
  
  Do you have a similar question?
  
  Recalculate according to your conditions!
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math problem with a detailed solution. It presents two sequences and asks for the next two terms, a recurrence relation, and an explicit formula for each sequence. The solution is provided for both sequences, including the calculations and formulas.
  
  Label: question-and-answer