



"""simple docstring"""
def   A_    (     _lowercase      ):







						'''simple docstring'''


						if not isinstance(_lowercase,   _lowercase      ):
												raise ValueError("""Input series is not valid, valid series - [2, 4, 8]"""      )
						if len(_lowercase      ) == 0:
												raise ValueError("""Input list must be a non empty list"""      )
						if len(_lowercase      ) == 1:
												return True
						try:
												snake_case_					:Optional[Any]		=     series[1] / series[0]
												for index in range(len(_lowercase      ) - 1      ):
																		if series[index + 1] / series[index] != common_ratio:
																								return False
						except ZeroDivisionError:
												return False
						return True
def   A_    (     _lowercase      ):







						'''simple docstring'''


						if not isinstance(_lowercase,   _lowercase      ):
												raise ValueError("""Input series is not valid, valid series - [2, 4, 8]"""      )
						if len(_lowercase      ) == 0:
												raise ValueError("""Input list must be a non empty list"""      )
						snake_case_					:Tuple		=     1
						for value in series:
												answer *= value
						return pow(_lowercase,   1 / len(_lowercase      )      )


if __name__ == "__main__":
						import doctest

						doctest.testmod()






def   A		(__A  :       int			)  -> int:







     """simple docstring"""


     if not isinstance(__A  ,							__A			):
          UpperCAmelCase_		=				F"""Input value of [number={number}] must be an integer"""
          raise TypeError(__A			)
     if number < 1:
          raise ValueError('''Input must be a positive integer'''			)
     return number * (2 * number - 1)


if __name__ == "__main__":
    import doctest

    doctest.testmod()






