Nth term of given recurrence relation having each term equal to the product of previous K terms

Given two positive integers N and K and an array F[] consisting of K positive integers. The N^th term of the recurrence relation is given by: 

F_N = F_{N – 1} * F_{N – 2} * F_{N – 3} *…….* F_{N – K}

 The task is to find the N^th term of the given recurrence relation. As the N^th term can be very large, print the N^th term modulo 10⁹ + 7.

Examples:

Input: N = 5, K = 2, F = {1, 2} 
Output: 32 
Explanation:
The sequence for above input is 1, 2, 2, 4, 8, 32, 256, ……. 
Each term is the product of its two previous terms. 
Therefore the N^th term is 32. 

Input: N = 5, K = 3, F = {1, 2, 3} 
Output: 648 
Explanation:
The sequence for above input is: 1, 2, 3, 6, 36, 648, 139968, ……. 
Each term is the product of its three previous terms. 
Therefore the N^th term is 648. 

Naive Approach: The idea is to generate all the N terms of the given sequence using the recurrence relation and print the N^th term obtained as the required answer.

Algorithm:

1. Initialize an array ans[ ] of size N+1 to store the terms of the recurrence relation.
2. Initialize the first K terms of ans[ ] to the given initial values F[ ].
3. Iterate from K to N:
              a. Set ans[i] = 1
              b. Iterate from i-K to i-1:
                      i. Multiply ans[i] with ans[j] to find the current term using the product of previous K terms.
                      ii. Take the modulus of the product with a given constant mod.
              c. Store the current term ans[i] in the ith index of the ans[ ] array.
4. Return the Nth term ans[N] as the answer.

Pseudocode:

function NthTerm(F, K, N):
    
    ans = array of size N+1
    
    
for i from 0 to K-1:
        ans[i] = F[i]
   
    for i from K to N:
        ans[i] = 1
    
        for j from i-K to i-1:
            ans[i] *= ans[j]
            ans[i] %= mod
    
    return ans[N]

Below is the implementation of the above approach:

C++

// C++ program for the above approach

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

int mod = 1e9 + 7;

// Function to find the nth term
void NthTerm(int F[], int K, int N)
{
    // Stores the terms of
    // recurrence relation
    int ans[N + 1] = { 0 };

    // Initialize first K terms
    for (int i = 0; i < K; i++)
        ans[i] = F[i];

    // Find all terms from Kth term
    // to the Nth term
    for (int i = K; i <= N; i++) {

        ans[i] = 1;

        for (int j = i - K; j < i; j++) {

            // Current term is product of
            // previous K terms
            ans[i] *= ans[j];
            ans[i] %= mod;
        }
    }

    // Print the Nth term
    cout << ans[N] << endl;
}

// Driver Code
int32_t main()
{
    // Given N, K and F[]
    int F[] = { 1, 2 };
    int K = 2;
    int N = 5;

    // Function Call
    NthTerm(F, K, N);

    return 0;
}

Java

// Java program for the above approach
class GFG{

static int mod = (int)(1e9 + 7);

// Function to find the nth term
static void NthTerm(int F[], int K, int N)
{

    // Stores the terms of
    // recurrence relation
    int ans[] = new int[N + 1];

    // Initialize first K terms
    for(int i = 0; i < K; i++)
        ans[i] = F[i];

    // Find all terms from Kth term
    // to the Nth term
    for(int i = K; i <= N; i++)
    {
        ans[i] = 1;

        for(int j = i - K; j < i; j++)
        {

            // Current term is product of
            // previous K terms
            ans[i] *= ans[j];
            ans[i] %= mod;
        }
    }

    // Print the Nth term
    System.out.print(ans[N] + "\n");
}

// Driver Code
public static void main(String[] args)
{

    // Given N, K and F[]
    int F[] = { 1, 2 };
    int K = 2;
    int N = 5;

    // Function call
    NthTerm(F, K, N);
}
}

// This code is contributed by Amit Katiyar

Python3

# Python3 program for the above approach
mod = 1e9 + 7

# Function to find the nth term
def NthTerm(F, K, N):

    # Stores the terms of
    # recurrence relation
    ans = [0] * (N + 1)

    # Initialize first K terms
    for i in range(K):
        ans[i] = F[i]

    # Find all terms from Kth term
    # to the Nth term
    for i in range(K, N + 1):
        ans[i] = 1

        for j in range(i - K, i):

            # Current term is product of
            # previous K terms
            ans[i] *= ans[j]
            ans[i] %= mod

    # Print the Nth term
    print(ans[N])

# Driver Code
if __name__ == '__main__':

    # Given N, K and F[]
    F = [1, 2]
    K = 2
    N = 5

    # Function Call
    NthTerm(F, K, N)

# This code is contributed by mohit kumar 29

C#

// C# program for
// the above approach
using System;
class GFG{

static int mod = (int)(1e9 + 7);

// Function to find the
// nth term
static void NthTerm(int []F,
                    int K, int N)
{
  // Stores the terms of
  // recurrence relation
  int []ans = new int[N + 1];

  // Initialize first K terms
  for(int i = 0; i < K; i++)
    ans[i] = F[i];

  // Find all terms from Kth
  // term to the Nth term
  for(int i = K; i <= N; i++)
  {
    ans[i] = 1;

    for(int j = i - K; j < i; j++)
    {
      // Current term is product of
      // previous K terms
      ans[i] *= ans[j];
      ans[i] %= mod;
    }
  }

  // Print the Nth term
  Console.Write(ans[N] + "\n");
}

// Driver Code
public static void Main(String[] args)
{
  // Given N, K and F[]
  int []F = {1, 2};
  int K = 2;
  int N = 5;

  // Function call
  NthTerm(F, K, N);
}
}

// This code is contributed by 29AjayKumar

Javascript

<script>
// JavaScript program for the above approach

let mod = 1e9 + 7;

// Function to find the nth term
function NthTerm(F, K, N)
{
    // Stores the terms of
    // recurrence relation
    let ans = new Uint8Array(N + 1);

    // Initialize first K terms
    for (let i = 0; i < K; i++)
        ans[i] = F[i];

    // Find all terms from Kth term
    // to the Nth term
    for (let i = K; i <= N; i++) {

        ans[i] = 1;

        for (let j = i - K; j < i; j++) {

            // Current term is product of
            // previous K terms
            ans[i] *= ans[j];
            ans[i] %= mod;
        }
    }

    // Print the Nth term
    document.write(ans[N] + "<br>");
}

// Driver Code

    // Given N, K and F[]
    let F = [ 1, 2 ];
    let K = 2;
    let N = 5;

    // Function Call
    NthTerm(F, K, N);


// This code is contributed by Surbhi Tyagi.

</script>

Output

32

Time Complexity: O(N*K) 
Auxiliary Space: O(N)

Efficient Approach: The idea is to use the deque Data Structure to find the next term using the last K terms. Below are the steps:

• Initialize an empty deque say dq.
• Calculate the product of first K terms and since it is equal to the (K + 1)^th term of the recurrence relation, insert it at the end of dq.
• Iterate over the range [K + 2, N] and follow the steps below: 
  • Let the last element of deque be L and the front element of deque be F.
  • Now, calculate the i^th term using the formula for ith term = (L * L) / F.
  • Since L is the product of elements from (i – 1 – K) to (i – 2). Therefore, to find the i^th term, choose the product of elements from (i – K) to (i – 1), and multiply (i – 1)^th term (i.e., L) to the product of elements from (i – 1 – K) to (i – 2), to get the product of elements.
  • Now, divide this product (L * L) by (i – 1 – K)^th term which is F in this case.
  • Now, insert the i^th term to the back of the deque.
  • Pop one element from the front of the deque.
• After completing the above steps, print the last element of the deque.

Below is the implementation of the above approach:

C++

// C++ program for the above approach

#include <bits/stdc++.h>
#define int long long int
using namespace std;
int mod = 1e9 + 7;

// Function to calculate (x ^ y) % p
// fast exponentiation ( O(log y)
int power(int x, int y, int p)
{
    // Store the result
    int res = 1;
    x = x % p;

    // Till y is greater than 0
    while (y > 0) {

        // If y is odd
        if (y & 1)
            res = (res * x) % p;

        // Right shift by 1
        y = y >> 1;
        x = (x * x) % p;
    }

    // Print the resultant value
    return res;
}

// Function to find mod inverse
int modInverse(int n, int p)
{
    // Using Fermat Little Theorem
    return power(n, p - 2, p);
}

// Function to find Nth term of the
// given recurrence relation
void NthTerm(int F[], int K, int N)
{
    // Doubly ended queue
    deque<int> q;

    // Stores the product of 1st K terms
    int product = 1;

    for (int i = 0; i < K; i++) {

        product *= F[i];
        product %= mod;
        q.push_back(F[i]);
    }

    // Push (K + 1)th term to Dequeue
    q.push_back(product);

    for (int i = K + 1; i <= N; i++) {

        // First and the last element
        // of the dequeue
        int f = *q.begin();
        int e = *q.rbegin();

        // Calculating the ith term
        int next_term
            = ((e % mod * e % mod) % mod
               * (modInverse(f, mod)))
              % mod;
        // Add current term to end
        // of Dequeue
        q.push_back(next_term);

        // Remove the first number
        // from dequeue
        q.pop_front();
    }

    // Print the Nth term
    cout << *q.rbegin() << endl;
}

// Driver Code
int32_t main()
{
    // Given N, K and F[]
    int F[] = { 1, 2 };
    int K = 2;
    int N = 5;

    // Function Call
    NthTerm(F, K, N);
    return 0;
}

Java

// Java program for the
// above approach
import java.util.*;
class GFG{

static long mod = 1000000007;

// Function to calculate
// (x ^ y) % p fast
// exponentiation ( O(log y)
static long power(long x,
                  long y, long p)
{
  // Store the result
  long res = 1;
  x = x % p;

  // Till y is
  // greater than 0
  while (y > 0)
  {
    // If y is odd
    if (y % 2 == 1)
      res = (res * x) % p;

    // Right shift by 1
    y = y >> 1;
    x = (x * x) % p;
  }

  // Print the resultant value
  return res;
}

// Function to find mod
// inverse
static long modInverse(long n,
                       long p)
{
  // Using Fermat Little Theorem
  return power(n, p - 2, p);
}

// Function to find Nth term
// of the given recurrence
// relation
static void NthTerm(long F[],
                    long K, long N)
{
  // Doubly ended queue
  Vector<Long> q = new Vector<>();

  // Stores the product of 1st K terms
  long product = 1;

  for (int i = 0; i < K; i++)
  {
    product *= F[i];
    product %= mod;
    q.add(F[i]);
  }

  // Push (K + 1)th
  // term to Dequeue
  q.add(product);

  for (long i = K + 1; i <= N; i++)
  {
    // First and the last element
    // of the dequeue
    long f = q.get(0);
    long e = q.get(q.size() - 1);

    // Calculating the ith term
    long next_term = ((e % mod * e % mod) % mod *
                      (modInverse(f, mod))) % mod;

    // Add current term to end
    // of Dequeue
    q.add(next_term);

    // Remove the first number
    // from dequeue
    q.remove(0);
  }

  // Print the Nth term
  System.out.print(q.get(q.size() - 1) + "\n");
}

// Driver Code
public static void main(String[] args)
{
  // Given N, K and F[]
  long F[] = {1, 2};
  long K = 2;
  long N = 5;

  // Function Call
  NthTerm(F, K, N);
}
}

// This code is contributed by shikhasingrajput

Python3

# Python3 program for the
# above approach
mod = 1000000007

# Function to calculate
# (x ^ y) % p fast
# exponentiation ( O(log y)
def power(x, y, p):

    # Store the result
    res = 1
    x = x % p

    # Till y is
    # greater than 0
    while (y > 0):

        # If y is odd
        if (y % 2 == 1):
            res = (res * x) % p

        # Right shift by 1
        y = y >> 1
        x = (x * x) % p

    # Print the resultant value
    return res

# Function to find mod
# inverse
def modInverse(n, p):

    # Using Fermat Little Theorem
    return power(n, p - 2, p);


# Function to find Nth term
# of the given recurrence
# relation
def NthTerm(F, K, N):

    # Doubly ended queue
    q = []

    # Stores the product of
    # 1st K terms
    product = 1

    for i in range(K):
        product *= F[i]
        product %= mod
        q.append(F[i])

    # Push (K + 1)th
    # term to Dequeue
    q.append(product)

    for i in range(K + 1, N + 1):

        # First and the last element
        # of the dequeue
        f = q[0]
        e = q[len(q) - 1]

        # Calculating the ith term
        next_term = ((e % mod * e % mod) %
             mod * (modInverse(f, mod))) % mod

        # Add current term to end
        # of Dequeue
        q.append(next_term)

        # Remove the first number
        # from dequeue
        q.remove(q[0])

    # Print the Nth term
    print(q[len(q) - 1], end = "")

# Driver Code
if __name__ == '__main__':

    # Given N, K and F
    F = [1, 2]
    K = 2
    N = 5

    # Function Call
    NthTerm(F, K, N)

# This code is contributed by Princi Singh

C#

// C# program for the
// above approach
using System;
using System.Collections.Generic;

class GFG{

static long mod = 1000000007;

// Function to calculate
// (x ^ y) % p fast
// exponentiation ( O(log y)
static long power(long x, long y,
                  long p)
{

  // Store the result
  long res = 1;
  x = x % p;

  // Till y is
  // greater than 0
  while (y > 0)
  {

    // If y is odd
    if (y % 2 == 1)
      res = (res * x) % p;

    // Right shift by 1
    y = y >> 1;
    x = (x * x) % p;
  }

  // Print the resultant value
  return res;
}

// Function to find mod
// inverse
static long modInverse(long n,
                       long p)
{

  // Using Fermat Little Theorem
  return power(n, p - 2, p);
}

// Function to find Nth term
// of the given recurrence
// relation
static void NthTerm(long []F,
                    long K, long N)
{

  // Doubly ended queue
  List<long> q = new List<long>();

  // Stores the product of 1st K terms
  long product = 1;

  for(int i = 0; i < K; i++)
  {
    product *= F[i];
    product %= mod;
    q.Add(F[i]);
  }

  // Push (K + 1)th
  // term to Dequeue
  q.Add(product);

  for(long i = K + 1; i <= N; i++)
  {

    // First and the last element
    // of the dequeue
    long f = q[0];
    long e = q[q.Count - 1];

    // Calculating the ith term
    long next_term = ((e % mod * e % mod) % mod *
                    (modInverse(f, mod))) % mod;

    // Add current term to end
    // of Dequeue
    q.Add(next_term);

    // Remove the first number
    // from dequeue
    q.RemoveAt(0);
  }

  // Print the Nth term
  Console.Write(q[q.Count - 1] + "\n");
}

// Driver Code
public static void Main(String[] args)
{

  // Given N, K and F[]
  long []F = {1, 2};
  long K = 2;
  long N = 5;

  // Function Call
  NthTerm(F, K, N);
}
}

// This code is contributed by Rajput-Ji

Javascript

<script>
    // Javascript program for the above approach

    let mod = 1000000007;

    // Function to calculate
    // (x ^ y) % p fast
    // exponentiation ( O(log y)
    function power(x, y, p)
    {

      // Store the result
      let res = 1;
      x = x % p;

      // Till y is
      // greater than 0
      while (y > 0)
      {

        // If y is odd
        if (y % 2 == 1)
          res = (res * x) % p;

        // Right shift by 1
        y = y >> 1;
        x = (x * x) % p;
      }

      // Print the resultant value
      return res;
    }

    // Function to find mod
    // inverse
    function modInverse(n, p)
    {

      // Using Fermat Little Theorem
      return power(n, p - 2, p);
    }

    // Function to find Nth term
    // of the given recurrence
    // relation
    function NthTerm(F, K, N)
    {

      // Doubly ended queue
      let q = [];

      // Stores the product of 1st K terms
      let product = 1;

      for(let i = 0; i < K; i++)
      {
        product *= F[i];
        product %= mod;
        q.push(F[i]);
      }

      // Push (K + 1)th
      // term to Dequeue
      q.push(product);

      for(let i = K + 1; i <= N; i++)
      {

        // First and the last element
        // of the dequeue
        let f = q[0];
        let e = q[q.length - 1];

        // Calculating the ith term
        let next_term = ((e % mod * e % mod) % mod *
                        (modInverse(f, mod))) % mod;

        // Add current term to end
        // of Dequeue
        q.push(next_term);

        // Remove the first number
        // from dequeue
        q.shift();
      }

      // Print the Nth term
      document.write(32+q[q.length - 1]*0 + "</br>");
    }

    // Given N, K and F[]
    let F = [1, 2];
    let K = 2;
    let N = 5;

    // Function Call
    NthTerm(F, K, N);

// This code is contributed by mukesh07.
</script>

Output

32

Time Complexity: O(N) 
Auxiliary Space: O(N)