💡 Problem Formulation: You’re given a sequence of numbers and you need to find the last digit of the nth entry in that sequence. Specifically, this article discusses how to determine the last digit for any given position n in a sequence, with n being a positive integer. For example, if the input sequence is a geometric progression like 2, 4, 8, 16, …, and n is 3, the desired output is 8 (the last digit of the third entry).
Method 1: Modulo Operation
This method involves using the modulo operation to obtain the last digit of the nth term in a sequence. In many sequences, especially those involving integers, the last digit repeats in a cycle. By calculating the nth term and then applying n % 10, you can find the last digit easily.
Here’s an example:
def last_digit_of_sequence(n, sequence_formula):
term = sequence_formula(n)
return term % 10
# Example for a sequence where nth term is 2**n
print(last_digit_of_sequence(3, lambda n: 2**n))The output will be 8.
The code snippet defines a function last_digit_of_sequence, that takes n and a lambda function representing the sequence formula. The lambda function calculates the nth term, and the modulo operation yields the last digit. This is a straightforward and efficient method for sequences with a clear mathematical formulation.
Method 2: String Conversion
This method converts the calculated term to a string and then picks the last character. This is particularly useful if you’re dealing with a sequence generated by an algorithm rather than a direct formula.
Here’s an example:
def last_digit_via_string(n, sequence_generator):
term = str(sequence_generator(n))
return int(term[-1])
# Example for Fibonacci sequence
def fibonacci(n):
a, b = 0, 1
for _ in range(n-1):
a, b = b, a+b
return b
print(last_digit_via_string(7, fibonacci))The output will be 3.
The function last_digit_via_string accepts n and a sequence generator function. It computes the nth term using the generator, converts it to a string, and then returns the last character as an integer. It’s simple, but might not be as efficient for large numbers because of string conversion overhead.
Method 3: Mathematical Pattern Observation
For some sequences, such as the powers of a number, the last digits repeat after a certain point. By observing these patterns, you can find the last digit directly without computation of the term.
Here’s an example:
def last_digit_pattern(n):
# Last digits of powers of 2 repeat every 4: 2, 4, 8, 6...
pattern = [2, 4, 8, 6]
return pattern[(n-1) % 4]
print(last_digit_pattern(3))The output will be 8.
The method relies on the fact that the last digits of powers of two cycle through 2, 4, 8, 6… The function last_digit_pattern maps n to the corresponding digit in this cycle. This method is extremely fast but applicable when the pattern is known and consistent.
Method 4: Recursive Function with Modulo
If a sequence can be generated recursively, with each term based on the previous one(s), you can incorporate the modulo operation into the recursion. This can avoid large number computations by keeping track of only the last digits throughout.
Here’s an example:
def recursive_last_digit(n):
if n <= 1:
return n
else:
return (recursive_last_digit(n-1) + recursive_last_digit(n-2)) % 10
print(recursive_last_digit(7))The output will be 3.
This code snippet utilizes a recursive function to compute the Fibonacci sequence, directly computing the last digit by using the modulo operation within the recursion. This keeps the numbers small, but the method is inefficient due to the exponential complexity of the naive recursion.
Bonus One-Liner Method 5: List Comprehension with Modulo
For sequences where you can generate a list up to n, you can use a one-liner list comprehension with modulo to get the last digit.
Here’s an example:
last_digit_list_comp = lambda n: [2**x % 10 for x in range(1, n+1)][-1] print(last_digit_list_comp(3))
The output will be 8.
This one-liner creates a list of the last digits of the powers of 2 up to n and then selects the last item. It’s clean and Pythonic but generates the entire list, which can be memory-intensive for large n.
Summary/Discussion
- Method 1: Modulo Operation. Easy to implement and efficient for sequences with direct mathematical formulas. Not suitable for sequences without simple formulas.
- Method 2: String Conversion. Good for algorithmically generated sequences. Less efficient due to string operations.
- Method 3: Mathematical Pattern Observation. The fastest method, but limited to sequences where a last digit pattern exists and is known.
- Method 4: Recursive Function with Modulo. Keeps numbers small throughout computation. Inefficient due to the high computational cost of recursion.
- Method 5: List Comprehension with Modulo. Pythonic and concise. Can be impractical for large sequences due to memory constraints.