# Sum of Fibonacci Numbers with alternate negatives

Given a positive integer n, the task is to find the value of F1 – F2 + F3 -……….+ (-1)n+1Fn where Fi denotes i-th Fibonacci number.

Fibonacci Numbers: The Fibonacci numbers are the numbers in the following integer sequence.

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

In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation

`Fn = Fn-1 + Fn-2`

with seed values F0 = 0 and F1 = 1.

Examples

```Input: n = 5
Output: 4
Explanation: 1 - 1 + 2 - 3 + 5 = 4

Input: n = 8
Output: -12
Explanation: 1 - 1 + 2 - 3 + 5 - 8 + 13 - 21 =  -12
```

Method 1: (O(n) time Complexity) This method includes solving the problem directly by finding all Fibonacci numbers till n and adding up the alternating sum. But this will require O(n) time complexity.

Below is the implementation of the above approach:

## C++

 `// C++ Program to find alternate sum ` `// of Fibonacci numbers ` ` `  `#include ` `using` `namespace` `std; ` ` `  `// Computes value of first fibonacci numbers ` `// and stores their alternate sum ` `int` `calculateAlternateSum(``int` `n) ` `{ ` `    ``if` `(n <= 0) ` `        ``return` `0; ` ` `  `    ``int` `fibo[n + 1]; ` `    ``fibo = 0, fibo = 1; ` ` `  `    ``// Initialize result ` `    ``int` `sum = ``pow``(fibo, 2) + ``pow``(fibo, 2); ` ` `  `    ``// Add remaining terms ` `    ``for` `(``int` `i = 2; i <= n; i++) { ` `        ``fibo[i] = fibo[i - 1] + fibo[i - 2]; ` ` `  `        ``// For even terms ` `        ``if` `(i % 2 == 0) ` `            ``sum -= fibo[i]; ` ` `  `        ``// For odd terms ` `        ``else` `            ``sum += fibo[i]; ` `    ``} ` ` `  `    ``// Return the alternating sum ` `    ``return` `sum; ` `} ` ` `  `// Driver program to test above function ` `int` `main() ` `{ ` ` `  `    ``// Get n ` `    ``int` `n = 8; ` ` `  `    ``// Find the alternating sum ` `    ``cout << ``"Alternating Fibonacci Sum upto "` `         ``<< n << ``" terms: "` `         ``<< calculateAlternateSum(n) << endl; ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java Program to find alternate sum  ` `// of Fibonacci numbers  ` ` `  `public` `class` `GFG { ` `     `  `    ``//Computes value of first fibonacci numbers  ` `    ``// and stores their alternate sum  ` `    ``static` `double` `calculateAlternateSum(``int` `n)  ` `    ``{  ` `        ``if` `(n <= ``0``)  ` `            ``return` `0``;  ` `       `  `        ``int` `fibo[] = ``new` `int` `[n + ``1``];  ` `        ``fibo[``0``] = ``0``; ` `        ``fibo[``1``] = ``1``;  ` `       `  `        ``// Initialize result  ` `        ``double` `sum = Math.pow(fibo[``0``], ``2``) + Math.pow(fibo[``1``], ``2``);  ` `       `  `        ``// Add remaining terms  ` `        ``for` `(``int` `i = ``2``; i <= n; i++) {  ` `            ``fibo[i] = fibo[i - ``1``] + fibo[i - ``2``];  ` `       `  `            ``// For even terms  ` `            ``if` `(i % ``2` `== ``0``)  ` `                ``sum -= fibo[i];  ` `       `  `            ``// For odd terms  ` `            ``else` `                ``sum += fibo[i];  ` `        ``}  ` `       `  `        ``// Return the alternating sum  ` `        ``return` `sum;  ` `    ``}  ` `       `  `     `  `    ``// Driver code ` `    ``public` `static` `void` `main(String args[]) ` `    ``{ ` `        ``// Get n  ` `        ``int` `n = ``8``;  ` `       `  `        ``// Find the alternating sum  ` `        ``System.out.println(``"Alternating Fibonacci Sum upto "` `              ``+ n + ``" terms: "` `              ``+ calculateAlternateSum(n));  ` `       `  `    ``} ` `    ``// This Code is contributed by ANKITRAI1 ` `} `

## Python 3

 `# Python 3 Program to find alternate sum ` `# of Fibonacci numbers ` ` `  `# Computes value of first fibonacci numbers ` `# and stores their alternate sum ` `def` `calculateAlternateSum(n): ` ` `  `    ``if` `(n <``=` `0``): ` `        ``return` `0` ` `  `    ``fibo ``=` `[``0``]``*``(n ``+` `1``) ` `    ``fibo[``0``] ``=` `0` `    ``fibo[``1``] ``=` `1` ` `  `    ``# Initialize result ` `    ``sum` `=` `pow``(fibo[``0``], ``2``) ``+` `pow``(fibo[``1``], ``2``) ` ` `  `    ``# Add remaining terms ` `    ``for` `i ``in` `range``(``2``, n``+``1``) : ` `        ``fibo[i] ``=` `fibo[i ``-` `1``] ``+` `fibo[i ``-` `2``] ` ` `  `        ``# For even terms ` `        ``if` `(i ``%` `2` `=``=` `0``): ` `            ``sum` `-``=` `fibo[i] ` ` `  `        ``# For odd terms ` `        ``else``: ` `            ``sum` `+``=` `fibo[i] ` ` `  `    ``# Return the alternating sum ` `    ``return` `sum` ` `  `# Driver program to test above function ` `if` `__name__ ``=``=` `"__main__"``: ` `    ``# Get n ` `    ``n ``=` `8` ` `  `    ``# Find the alternating sum ` `    ``print``( ``"Alternating Fibonacci Sum upto "` `        ``, n ,``" terms: "` `        ``, calculateAlternateSum(n)) ` ` `  `# this code is contributed by ` `# ChitraNayal `

## C#

 `// C# Program to find alternate sum  ` `// of Fibonacci numbers  ` `using` `System; ` ` `  `class` `GFG  ` `{ ` ` `  `// Computes value of first fibonacci numbers  ` `// and stores their alternate sum  ` `static` `double` `calculateAlternateSum(``int` `n)  ` `{  ` `    ``if` `(n <= 0)  ` `        ``return` `0;  ` ` `  `    ``int` `[]fibo = ``new` `int` `[n + 1];  ` `    ``fibo = 0; ` `    ``fibo = 1;  ` ` `  `    ``// Initialize result  ` `    ``double` `sum = Math.Pow(fibo, 2) +  ` `                 ``Math.Pow(fibo, 2);  ` ` `  `    ``// Add remaining terms  ` `    ``for` `(``int` `i = 2; i <= n; i++)  ` `    ``{  ` `        ``fibo[i] = fibo[i - 1] + fibo[i - 2];  ` ` `  `        ``// For even terms  ` `        ``if` `(i % 2 == 0)  ` `            ``sum -= fibo[i];  ` ` `  `        ``// For odd terms  ` `        ``else` `            ``sum += fibo[i];  ` `    ``}  ` ` `  `    ``// Return the alternating sum  ` `    ``return` `sum;  ` `}  ` ` `  `// Driver code ` `public` `static` `void` `Main() ` `{ ` `    ``// Get n  ` `    ``int` `n = 8;  ` ` `  `    ``// Find the alternating sum  ` `    ``Console.WriteLine(``"Alternating Fibonacci Sum upto "` `+  ` `              ``n + ``" terms: "` `+ calculateAlternateSum(n));  ` ` `  `} ` `} ` ` `  `// This code is contributed by inder_verma `

## PHP

 ` `

Output:

```Alternating Fibonacci Sum upto 8 terms: -12
```

Method 2: (O(log n) Complexity) This method involves the following observation to reduce the time complexity:

• For n = 2,
F1 – F2 = 1 – 1
= 0
= 1 + (-1)3 * F1
• For n = 3,
F1 – F2 + F3
= 1 – 1 + 2
= 2
= 1 + (-1)4 * F2
• For n = 4,
F1 – F2 + F3 – F4
= 1 – 1 + 2 – 3
= -1
= 1 + (-1)5 * F3
• For n = m,
F1 – F2 + F3 -…….+ (-1)m+1 * Fm-1
= 1 + (-1)m+1Fm-1

Assuming this to be true. Now if (n = m+1) is also true, it means that the assumption is correct. Otherwise it is wrong.

• For n = m+1,
F1 – F2 + F3 -…….+ (-1)m+1 * Fm + (-1)m+2 * Fm+1
= 1 + (-1)m+1 * Fm-1 + (-1)m+2 * Fm+1
= 1 + (-1)m+1(Fm-1 – Fm+1)
= 1 + (-1)m+1(-Fm) = 1 + (-1)m+2(Fm)

which is true as per the assumption for n = m.

Hence general term for the alternating Fibonacci Sum:

F1 – F2 + F3 -…….+ (-1)n+1 Fn = 1 + (-1)n+1Fn-1

So in order to find alternate sum, only the n-th Fibonacci term is to be found, which can be done in O(log n) time( Refer to Method 5 or 6 of this article.)

Below is the implementation of method 6 of this:

## C++

 `// C++ Program to find alternate  Fibonacci Sum in ` `// O(Log n) time. ` ` `  `#include ` `using` `namespace` `std; ` ` `  `const` `int` `MAX = 1000; ` ` `  `// Create an array for memoization ` `int` `f[MAX] = { 0 }; ` ` `  `// Returns n'th Fibonacci number ` `// using table f[] ` `int` `fib(``int` `n) ` `{ ` `    ``// Base cases ` `    ``if` `(n == 0) ` `        ``return` `0; ` `    ``if` `(n == 1 || n == 2) ` `        ``return` `(f[n] = 1); ` ` `  `    ``// If fib(n) is already computed ` `    ``if` `(f[n]) ` `        ``return` `f[n]; ` ` `  `    ``int` `k = (n & 1) ? (n + 1) / 2 : n / 2; ` ` `  `    ``// Applying above formula [Note value n&1 is 1 ` `    ``// if n is odd, else 0]. ` `    ``f[n] = (n & 1) ? (fib(k) * fib(k) + fib(k - 1) * fib(k - 1)) ` `                   ``: (2 * fib(k - 1) + fib(k)) * fib(k); ` ` `  `    ``return` `f[n]; ` `} ` ` `  `// Computes value of Alternate  Fibonacci Sum ` `int` `calculateAlternateSum(``int` `n) ` `{ ` `    ``if` `(n % 2 == 0) ` `        ``return` `(1 - fib(n - 1)); ` `    ``else` `        ``return` `(1 + fib(n - 1)); ` `} ` ` `  `// Driver program to test above function ` `int` `main() ` `{ ` `    ``// Get n ` `    ``int` `n = 8; ` ` `  `    ``// Find the alternating sum ` `    ``cout << ``"Alternating Fibonacci Sum upto "` `         ``<< n << ``" terms  : "` `         ``<< calculateAlternateSum(n) << endl; ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java Program to find alternate  ` `// Fibonacci Sum in O(Log n) time. ` `class` `GFG { ` ` `  `    ``static` `final` `int` `MAX = ``1000``; ` ` `  `    ``// Create an array for memoization  ` `    ``static` `int` `f[] = ``new` `int``[MAX]; ` ` `  `    ``// Returns n'th Fibonacci number  ` `    ``// using table f[]  ` `    ``static` `int` `fib(``int` `n) { ` `         `  `        ``// Base cases  ` `        ``if` `(n == ``0``) { ` `            ``return` `0``; ` `        ``} ` `        ``if` `(n == ``1` `|| n == ``2``) { ` `            ``return` `(f[n] = ``1``); ` `        ``} ` ` `  `        ``// If fib(n) is already computed  ` `        ``if` `(f[n] > ``0``) { ` `            ``return` `f[n]; ` `        ``} ` ` `  `        ``int` `k = (n % ``2` `== ``1``) ? (n + ``1``) / ``2` `: n / ``2``; ` ` `  `        ``// Applying above formula [Note value n&1 is 1  ` `        ``// if n is odd, else 0].  ` `        ``f[n] = (n % ``2` `== ``1``) ? (fib(k) * fib(k) + fib(k - ``1``) * fib(k - ``1``)) ` `                ``: (``2` `* fib(k - ``1``) + fib(k)) * fib(k); ` ` `  `        ``return` `f[n]; ` `    ``} ` ` `  `    ``// Computes value of Alternate Fibonacci Sum  ` `    ``static` `int` `calculateAlternateSum(``int` `n) { ` `        ``if` `(n % ``2` `== ``0``) { ` `            ``return` `(``1` `- fib(n - ``1``)); ` `        ``} ``else` `{ ` `            ``return` `(``1` `+ fib(n - ``1``)); ` `        ``} ` `    ``} ` ` `  `// Driver program to test above function  ` `    ``public` `static` `void` `main(String[] args) { ` `        ``// Get n  ` `        ``int` `n = ``8``; ` ` `  `        ``// Find the alternating sum  ` `        ``System.out.println(``"Alternating Fibonacci Sum upto "` `                ``+ n + ``" terms : "` `                ``+ calculateAlternateSum(n)); ` `    ``} ` `} ` `// This code is contributed by PrinciRaj1992 `

## C#

 `// C# Program to find alternate  ` `// Fibonacci Sum in O(Log n) time. ` `using` `System; ` ` `  `class` `GFG  ` `{ ` `    ``static` `readonly` `int` `MAX = 1000; ` ` `  `    ``// Create an array for memoization  ` `    ``static` `int` `[]f = ``new` `int``[MAX]; ` ` `  `    ``// Returns n'th Fibonacci number  ` `    ``// using table f[]  ` `    ``static` `int` `fib(``int` `n) ` `    ``{ ` `        ``// Base cases  ` `        ``if` `(n == 0)  ` `        ``{ ` `            ``return` `0; ` `        ``} ` `        ``if` `(n == 1 || n == 2)  ` `        ``{ ` `            ``return` `(f[n] = 1); ` `        ``} ` ` `  `        ``// If fib(n) is already computed  ` `        ``if` `(f[n] > 0)  ` `        ``{ ` `            ``return` `f[n]; ` `        ``} ` ` `  `        ``int` `k = (n % 2 == 1) ? (n + 1) / 2 : n / 2; ` ` `  `        ``// Applying above formula [Note value n&1 is 1  ` `        ``// if n is odd, else 0].  ` `        ``f[n] = (n % 2 == 1) ? (fib(k) * fib(k) +  ` `                        ``fib(k - 1) * fib(k - 1)) ` `                        ``: (2 * fib(k - 1) + fib(k)) * fib(k); ` ` `  `        ``return` `f[n]; ` `    ``} ` ` `  `    ``// Computes value of Alternate Fibonacci Sum  ` `    ``static` `int` `calculateAlternateSum(``int` `n)  ` `    ``{ ` `        ``if` `(n % 2 == 0)  ` `        ``{ ` `            ``return` `(1 - fib(n - 1)); ` `        ``} ``else`  `        ``{ ` `            ``return` `(1 + fib(n - 1)); ` `        ``} ` `    ``} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `Main()  ` `    ``{ ` `        ``// Get n  ` `        ``int` `n = 8; ` `         `  `        ``// Find the alternating sum  ` `        ``Console.WriteLine(``"Alternating Fibonacci Sum upto "` `                ``+ n + ``" terms : "` `                ``+ calculateAlternateSum(n)); ` `    ``} ` `} ` ` `  `// This code is contributed by 29AjayKumar `

Output:

```Alternating Fibonacci Sum upto 8 terms: -12
```

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