code: 
  
  
  def lowerCamelCase    (    lowerCAmelCase       :			list	):
  
  
  
  
  
  	"""simple docstring"""
  
  
  
  
  
  
  
  	if not isinstance(lowerCAmelCase	,	lowerCAmelCase	):
  		raise ValueError('Input series is not valid, valid series - [2, 4, 8]'	)
  	if len(lowerCAmelCase	) == 0:
  		raise ValueError('Input list must be a non empty list'	)
  	if len(lowerCAmelCase	) == 1:
  		return True
  	try:
  		__magic_name__ :					Optional[Any]  			=  series[1] / series[0]
  		for index in range(len(lowerCAmelCase	) - 1	):
  			if series[index + 1] / series[index] != common_ratio:
  				return False
  	except ZeroDivisionError:
  		return False
  	return True
  
  
  
  
  
  def lowerCamelCase    (    lowerCAmelCase       :			list	):
  
  
  
  
  
  	"""simple docstring"""
  
  
  
  
  
  
  
  	if not isinstance(lowerCAmelCase	,	lowerCAmelCase	):
  		raise ValueError('Input series is not valid, valid series - [2, 4, 8]'	)
  	if len(lowerCAmelCase	) == 0:
  		raise ValueError('Input list must be a non empty list'	)
  	__magic_name__ :					List[str]  			=  1
  	for value in series:
  		answer *= value
  	return pow(lowerCAmelCase	,	1 / len(lowerCAmelCase	)	)
  
  
  if __name__ == "__main__":
  							import doctest
  
  							doctest.testmod()
style_context: 
  
  
  '''simple docstring'''
  
  from __future__ import annotations
  
  
  
  
  
  def lowerCamelCase    (    lowerCAmelCase       :			list[int | float]	,	lowerCAmelCase       :			int	,	lowerCAmelCase       :			int	):
  
  
  
  
  
  	"""simple docstring"""
  
  
  
  
  
  
  
  	if len(lowerCAmelCase	) == 0:
  		raise ValueError('find_min() arg is an empty sequence'	)
  	if (
  	    left >= len(lowerCAmelCase	)
  	    or left < -len(lowerCAmelCase	)
  	    or right >= len(lowerCAmelCase	)
  	    or right < -len(lowerCAmelCase	)
  	):
  		raise IndexError('list index out of range'	)
  	if left == right:
  		return nums[left]
  	__magic_name__ :					Tuple  			=  (left + right) >> 1  # the middle
  	__magic_name__ :					str  			=  find_min(lowerCAmelCase	,	lowerCAmelCase	,	lowerCAmelCase	)  # find min in range[left, mid]
  	__magic_name__ :					Optional[Any]  			=  find_min(lowerCAmelCase	,	mid + 1	,	lowerCAmelCase	)  # find min in range[mid + 1, right]
  
  	return left_min if left_min <= right_min else right_min
  
  
  if __name__ == "__main__":
  							import doctest
  
  							doctest.testmod(verbose=True)