





"""simple docstring"""


def 				a__      (	_UpperCAmelCase						)     ->	int:

 '''simple docstring'''






 return n * (2 * n - 1)





def 				a__      (	_UpperCAmelCase						)     ->	bool:

 '''simple docstring'''






 snake_case__     				=					(1 + 2_4 * n) ** 0.5
 return ((1 + root) / 6) % 1 == 0





def 				a__      (	_UpperCAmelCase = 1_4_4						)     ->	int:

 '''simple docstring'''






 snake_case__     				=					start
 snake_case__     				=					hexagonal_num(_UpperCAmelCase						)
 while not is_pentagonal(_UpperCAmelCase						):
  n += 1
  snake_case__     				=					hexagonal_num(_UpperCAmelCase						)
 return num


if __name__ == "__main__":
   print(f'''{solution()} = ''')








"""simple docstring"""


def 				a__      (	_UpperCAmelCase						,  _UpperCAmelCase						,  _UpperCAmelCase						)     ->	float:

 '''simple docstring'''






 snake_case__     				=					[1 / first_term]
 snake_case__     				=					1 / first_term
 for _ in range(number_of_terms - 1						):
  first_term += common_difference
  arithmetic_progression.append(_UpperCAmelCase						)
 snake_case__     				=					[1 / step for step in arithmetic_progression]
 return sum(_UpperCAmelCase						)


if __name__ == "__main__":
   import doctest

   doctest.testmod()
   print(sum_of_harmonic_progression(1 / 2, 2, 2))

