Скачать или смотреть Finding the Last Digit of Partial Sum of the Fibonacci Series

Learn how to efficiently calculate the last digit of the sum of a segment of the Fibonacci series using the Pisano Period.
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Understanding the Problem: Last Digit of Partial Sum of Fibonacci Series

The Fibonacci sequence is a well-known series in mathematics where each number is the sum of the two preceding ones, often starting with 0 and 1. If you're tasked with finding the last digit of a sum of the Fibonacci numbers between two given indices, you're in for an interesting challenge!

The twist, however, is that when you consider the Fibonacci sequence modulo 10, the last digits repeat every 60 numbers. This sequence is called the Pisano period. Knowing this, we can devise a more efficient method to calculate the last digit of partial sums in the Fibonacci series.

The Key Insight: Pisano Period

Pisano Period: The last digits of Fibonacci numbers repeat every 60 terms when taken modulo 10. This means that instead of calculating a potentially large list of numbers, we only need the last 60 digits for computations.

Why 60?: This is a characteristic of Fibonacci numbers that comes from number theory and modular arithmetic.

Step-by-Step Solution

1. Precompute the Last Digits

First, we generate and store the last digits of the first 60 Fibonacci numbers. This allows us to easily access the required last digits without recalculating them every time.

2. Adjust the Range Using Modulo

To find the sum of a range from n to m, we can:

Calculate how many complete periods of 60 fit into this range.

Utilize these periods to speed up the calculation by summing them and only addressing the leftover segments outside these complete periods.

3. Implementing the Solution in C+ +

Here’s how you can structure the solution programmatically using C+ + :

[[See Video to Reveal this Text or Code Snippet]]

4. Understanding the Code

Array Initialization: The long long a[60] array is initialized to hold the last digits of the Fibonacci numbers up to the Pisano period.

Sum Calculation: The total last digit sum is accumulated while iterating through the first 60 Fibonacci numbers.

Counting Full Periods: The number of complete Pisano periods within your range is counted, and their contributions to the sum are calculated in one go.

Remainder Handling: Finally, any remaining terms are computed directly, ensuring an efficient calculation even for large ranges.

Conclusion

Calculating the last digit of a sum of Fibonacci numbers might seem daunting, especially for large ranges, but understanding the Pisano period allows us to formulate a much simpler and efficient approach. With the above method, leveraging the repeating pattern of last digits helps you get accurate results efficiently.

Now you can tackle this problem confidently and expand on it as needed for other applications within mathematics or computer science. Happy coding!