
def 		A						(  SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE							):




			"""simple docstring"""


			if start == end:
						return start if lst[start] > item else start + 1
			if start > end:
						return start

			UpperCAmelCase__		:Union[str, Any] 		=    (start + end) // 2
			if lst[mid] < item:
						return binary_search(SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE     ,						mid + 1     ,						SCREAMING_SNAKE_CASE							)
			elif lst[mid] > item:
						return binary_search(SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE     ,						mid - 1							)
			else:
						return mid




def 		A						(  SCREAMING_SNAKE_CASE							):




			"""simple docstring"""


			UpperCAmelCase__		:Optional[int] 		=    len(SCREAMING_SNAKE_CASE							)

			for index in range(1     ,						SCREAMING_SNAKE_CASE							):
						UpperCAmelCase__		:Optional[Any] 		=    lst[index]
						UpperCAmelCase__		:Tuple 		=    binary_search(SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE     ,						0     ,						index - 1							)
						UpperCAmelCase__		:Dict 		=    lst[:pos] + [value] + lst[pos:index] + lst[index + 1 :]

			return lst




def 		A						(  SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE							):




			"""simple docstring"""


			if not left:
						return right

			if not right:
						return left

			if left[0] < right[0]:
						return [left[0], *merge(left[1:]     ,						SCREAMING_SNAKE_CASE							)]

			return [right[0], *merge(SCREAMING_SNAKE_CASE     ,						right[1:]							)]




def 		A						(  SCREAMING_SNAKE_CASE							):




			"""simple docstring"""


			UpperCAmelCase__		:List[str] 		=    len(SCREAMING_SNAKE_CASE							)
			UpperCAmelCase__		,       UpperCAmelCase__		:List[Any] 		=    [], []
			UpperCAmelCase__		:Dict 		=    [lst[0]]
			UpperCAmelCase__		:Optional[int] 		=    []
			UpperCAmelCase__		:int 		=    1
			while i < length:
						if lst[i] < lst[i - 1]:
									runs.append(SCREAMING_SNAKE_CASE							)
									UpperCAmelCase__		:Dict 		=    [lst[i]]
						else:
									new_run.append(lst[i]							)
						i += 1
			runs.append(SCREAMING_SNAKE_CASE							)

			for run in runs:
						sorted_runs.append(insertion_sort(SCREAMING_SNAKE_CASE							)							)
			for run in sorted_runs:
						UpperCAmelCase__		:str 		=    merge(SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE							)

			return sorted_array




def 		A						(  ):




			"""simple docstring"""


			UpperCAmelCase__		:Union[str, Any] 		=    [5, 9, 10, 3, -4, 5, 178, 92, 46, -18, 0, 7]
			UpperCAmelCase__		:Optional[int] 		=    tim_sort(SCREAMING_SNAKE_CASE							)
			print(SCREAMING_SNAKE_CASE							)


if __name__ == "__main__":
						main()







from __future__ import annotations




def 		A						(  SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE     ,						):




			"""simple docstring"""


			if not all((nth_term, start_term_a, common_ratio_r)							):
						return []
			UpperCAmelCase__		:list[float | int] 		=    []
			UpperCAmelCase__		:str 		=    1
			UpperCAmelCase__		:Any 		=    common_ratio_r
			for _ in range(int(SCREAMING_SNAKE_CASE							)							):
						if not series:
									series.append(SCREAMING_SNAKE_CASE							)
						else:
									power += 1
									series.append(float(start_term_a * multiple							)							)
									UpperCAmelCase__		:str 		=    pow(float(SCREAMING_SNAKE_CASE							)     ,						SCREAMING_SNAKE_CASE							)
			return series


if __name__ == "__main__":
						import doctest

						doctest.testmod()

						__snake_case					:							Optional[int]           =							float(input('Enter the last number (n term) of the Geometric Series'))
						__snake_case					:							List[str]           =							float(input('Enter the starting term (a) of the Geometric Series'))
						__snake_case					:							Optional[int]           =							float(
						    input('Enter the common ratio between two terms (r) of the Geometric Series')
						)
						print('Formula of Geometric Series => a + ar + ar^2 ... +ar^n')
						print(geometric_series(nth_term, start_term_a, common_ratio_r))





