Skip to content
Related Articles
Sum of numbers in the Kth level of a Fibonacci triangle
• Last Updated : 23 Apr, 2021

Given a number K, the task is to find the sum of numbers at the Kth level of the Fibonacci triangle.
Examples:

```Input: K = 3
Output: 10
Explanation:
Fibonacci triangle till level 3:
0
1 1
2 3 5
Sum at 3rd level = 2 + 3 + 5 = 10

Input: K = 2
Output: 2
Explanation:
Fibonacci triangle till level 3:
0
1 1
Sum at 3rd level = 1 + 1 = 2 ```

Approach:

1. Till Kth level, i.e. from level [1, K-1], count of Fibonacci numbers already used can be computed as:

```cnt = N(Level 1) + N(Level 2)
+ N(Level 3) + ...
+ N(Level K-1)
= 1 + 2 + 3 + ... + (K-1)
= K*(K-1)/2```
1.
2. Also, we know that the Kth level will contain K Fibonacci numbers.
3. Therfore we can find the numbers in the Kth level as Fibonacci numbers in the range [(cnt + 1), (cnt + 1 + K)].
4. We can find the sum of Fibonacci numbers in a range in O(1) time using Binet’s Formula.

Below is the implementation of the above approach:

## C++

 `// C++ implementation to find``// the Sum of numbers in the``// Kth level of a Fibonacci triangle` `#include ``using` `namespace` `std;``#define MAX 1000000` `// Function to return``// the nth Fibonacci number``int` `fib(``int` `n)``{``    ``double` `phi = (1 + ``sqrt``(5)) / 2;``    ``return` `round(``pow``(phi, n) / ``sqrt``(5));``}` `// Function to return``// the required sum of the array``int` `calculateSum(``int` `l, ``int` `r)``{` `    ``// Using our deduced result``    ``int` `sum = fib(r + 2) - fib(l + 1);` `    ``return` `sum;``}` `// Function to return the sum of``// fibonacci in the Kth array``int` `sumFibonacci(``int` `k)``{``    ``// Count of fibonacci which are in``    ``// the arrays from 1 to k - 1``    ``int` `l = (k * (k - 1)) / 2;``    ``int` `r = l + k;` `    ``int` `sum = calculateSum(l, r - 1);` `    ``return` `sum;``}` `// Driver code``int` `main()``{` `    ``int` `k = 3;` `    ``cout << sumFibonacci(k);` `    ``return` `0;``}`

## Java

 `// Java implementation to find``// the Sum of numbers in the``// Kth level of a Fibonacci triangle``import` `java.util.*;` `class` `GFG``{` `// Function to return``// the nth Fibonacci number``static` `int` `fib(``int` `n)``{``    ``double` `phi = (``1` `+ Math.sqrt(``5``)) / ``2``;``    ``return` `(``int``)Math.round(Math.pow(phi, n) / Math.sqrt(``5``));``}` `// Function to return``// the required sum of the array``static` `int` `calculateSum(``int` `l, ``int` `r)``{` `    ``// Using our deduced result``    ``int` `sum = fib(r + ``2``) - fib(l + ``1``);` `    ``return` `sum;``}` `// Function to return the sum of``// fibonacci in the Kth array``static` `int` `sumFibonacci(``int` `k)``{``    ``// Count of fibonacci which are in``    ``// the arrays from 1 to k - 1``    ``int` `l = (k * (k - ``1``)) / ``2``;``    ``int` `r = l + k;` `    ``int` `sum = calculateSum(l, r - ``1``);` `    ``return` `sum;``}` `// Driver code``public` `static` `void` `main(String args[])``{` `    ``int` `k = ``3``;` `    ``System.out.println(sumFibonacci(k));``}``}` `// This code is contributed by AbhiThakur`

## Python3

 `# Python3 implementation to find``# the Sum of numbers in the``# Kth level of a Fibonacci triangle` `import` `math``MAX` `=` `1000000` `# Function to return``# the nth Fibonacci number``def` `fib(n):` `    ``phi ``=` `(``1` `+` `math.sqrt(``5``)) ``/` `2``    ``return` `round``(``pow``(phi, n) ``/` `math.sqrt(``5``))`` `  `# Function to return``# the required sum of the array``def` `calculateSum(l, r):` `    ``# Using our deduced result``    ``sum` `=` `fib(r ``+` `2``) ``-` `fib(l ``+` `1``)` `    ``return` `sum` `# Function to return the sum of``# fibonacci in the Kth array``def` `sumFibonacci(k) :``    ``# Count of fibonacci which are in``    ``# the arrays from 1 to k - 1``    ``l ``=` `(k ``*` `(k ``-` `1``)) ``/` `2``    ``r ``=` `l ``+` `k` `    ``sum` `=` `calculateSum(l, r ``-` `1``)` `    ``return` `sum` `# Driver code``k ``=` `3` `print``(sumFibonacci(k))` `# This code is contributed by Sanjit_Prasad`

## C#

 `// C# implementation to find``// the Sum of numbers in the``// Kth level of a Fibonacci triangle``using` `System;` `class` `GFG ``{``  ` `// Function to return``// the nth Fibonacci number``static` `int` `fib(``int` `n)``{``    ``double` `phi = (1 + Math.Sqrt(5)) / 2;``    ``return` `(``int``)Math.Round(Math.Pow(phi, n) / Math.Sqrt(5));``}`` ` `// Function to return``// the required sum of the array``static` `int` `calculateSum(``int` `l, ``int` `r)``{`` ` `    ``// Using our deduced result``    ``int` `sum = fib(r + 2) - fib(l + 1);`` ` `    ``return` `sum;``}`` ` `// Function to return the sum of``// fibonacci in the Kth array``static` `int` `sumFibonacci(``int` `k)``{``    ``// Count of fibonacci which are in``    ``// the arrays from 1 to k - 1``    ``int` `l = (k * (k - 1)) / 2;``    ``int` `r = l + k;`` ` `    ``int` `sum = calculateSum(l, r - 1);`` ` `    ``return` `sum;``}`` ` `// Driver code``public` `static` `void` `Main() ``{ `` ` `    ``int` `k = 3;`` ` `    ``Console.Write(sumFibonacci(k));``}``}` `// This code is contributed by mohit kumar 29`

## Javascript

 ``
Output:
`10`

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.  To complete your preparation from learning a language to DS Algo and many more,  please refer Complete Interview Preparation Course.

In case you wish to attend live classes with industry experts, please refer Geeks Classes Live

My Personal Notes arrow_drop_up