Last digit of sum of numbers in the given range in the Fibonacci series

Last Updated : 14 Feb, 2023

Given two non-negative integers M, N which signifies the range [M, N] where M ? N, the task is to find the last digit of the sum of F_M + F_M+1... + F_N where F_K is the K^th Fibonacci number in the Fibonacci series.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

Examples:

Input: M = 3, N = 9
Output: 6
Explanation:
We need to find F₃ + F₄ + F₅ + F₆ + F₇ + F₈ + F₉
=> 2 + 3 + 5 + 8 + 13 + 21 + 34 = 86.
Clearly, the last digit of the sum is 6.

Input: M = 3, N = 7
Output: 1
Explanation:
We need to find F₃ + F₄ + F₅ + F₆ + F₇
=> 2 + 3 + 5 + 8 + 13 = 31.
Clearly, the last digit of the sum is 1.

Naive Approach: The naive approach for this problem is to one by one find the sum of all K^th Fibonacci Numbers where K lies in the range [M, N] and return the last digit of the sum in the end. The time complexity for this approach is O(N) and this method fails for higher-ordered values of N.

Efficient approach:

C++

#include <bits/stdc++.h>
using namespace std;

long long getFibonacciPartialSumFast(long long m, long long n) {
    long long sum = 0;

    m = m % 60;
    n = n % 60;

    if (n < m) n += 60;

    long long current = 0;
    long long next = 1;

    for (int i = 0; i <= n; i++) {
        if (i >= m) {
            sum += current;
        }

        long long newCurrent = next;
        next = next + current;
        current = newCurrent;
    }

    return sum % 10;
}

int main() {
    long long m = 2;
    long long n = 10;
    cout << getFibonacciPartialSumFast(m, n) << endl;
    return 0;
}

Java

import java.util.*;

public class FibonacciPartialSum {
    private static long getFibonacciPartialSumFast(long m,
                                                   long n)
    {
        long sum = 0;

        // The input arguments, as the last digit
        // pattern repeats with a period of 60, and the sum
        // of 60 such consecutive numbers is 0 mod 10
      
        m = (int)(m % 60);
        n = (int)(n % 60);

        // Make sure n is greater than m
        if (n < m)
            n += 60;

        long current = 0;
        long next = 1;

        for (int i = 0; i <= n; ++i) {
            if (i >= m) {
                sum += current;
            }

            long newCurrent = next;
            next = next + current;
            current = newCurrent;
        }

        return (int)(sum % 10);
    }

    public static void main(String[] args)
    {
        Scanner scanner = new Scanner(System.in);
        long m = 2;
        long n = 10;
        System.out.println(
            getFibonacciPartialSumFast(m, n));
    }
}

Python3

def getFibonacciPartialSumFast(m, n):
    sum = 0

    # The input arguments, as the last digit
    # pattern repeats with a period of 60, and the sum
    # of 60 such consecutive numbers is 0 mod 10
    m = m % 60
    n = n % 60

    # Make sure n is greater than m
    if n < m:
        n += 60

    current = 0
    next = 1

    for i in range(n + 1):
        if i >= m:
            sum += current

        new_current = next
        next = next + current
        current = new_current

    return sum % 10

if __name__ == "__main__":
    m = 2
    n = 10
    print(getFibonacciPartialSumFast(m, n))

using System;
using System.Linq;
using System.Collections.Generic;

class GFG {

  static long getFibonacciPartialSumFast(long m, long n) {
    long sum = 0;

    // The input arguments, as the last digit
    // pattern repeats with a period of 60, and the sum
    // of 60 such consecutive numbers is 0 mod 10

    m = m % 60;
    n = n % 60;

    // Make sure n is greater than m
    if (n < m) 
      n += 60;

    long current = 0;
    long next = 1;

    for (int i = 0; i <= n; i++) {
      if (i >= m) {
        sum += current;
      }

      long newCurrent = next;
      next = next + current;
      current = newCurrent;
    }

    return sum % 10;
  }

  static public void Main()
  {
    long m = 2;
    long n = 10;
    Console.Write(getFibonacciPartialSumFast(m, n));
  }
}

// This code is contributed by ratiagrawal.

JavaScript

function getFibonacciPartialSumFast(m, n)
{
    let sum = 0;

    m = m % 60;
    n = n % 60;

    if (n < m) 
        n += 60;

    let current = 0;
    let next = 1;

    for (let i = 0; i <= n; i++) {
        if (i >= m) {
            sum += current;
        }

        let newCurrent = next;
        next = next + current;
        current = newCurrent;
    }

    return sum % 10;
}

let m = 2;
let n = 10;
console.log(getFibonacciPartialSumFast(m, n));

// This code is contributed by poojaagarwal2.

Output

K- Fibonacci series

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Last digit of sum of numbers in the given range in the Fibonacci series

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