Meaning of the formula how to find lost element in array?

Question

The task is to find lost element in the array. I understand the logic of the solution but I don't understand how does this formula works?

Here is the solution

int[] array = new int[]{4,1,2,3,5,8,6};
   int size = array.length;
   int result = (size + 1) * (size + 2)/2;
   for (int i : array){
       result -= i;
   }

But why we add 1 to total size and multiply it to total size + 2 /2 ?? In all resources, people just use that formula but nobody explains how that formula works

Answer 1

The sum of the digits 1 thru n is equal to ((n)(n+1))/2.

e.g. for 1,2,3,4,5 5*6/2 = 15.

But this is just a quick way to add up the numbers from 1 to n. Here is what is really going on.

The series computes the sum of 1 to n assuming they all were present. But by subtracting each number from that sum, the remainder is the missing number.

The formula for an arithmetic series of integers from k to n where adjacent elements differ by 1 is.

S_[k,n] = (n-k+1)(n+k)/2

Example: k = 5, n = 10

• S_[k,n] = 5 6 7 8 9 10

• S_[k,n] = 10 9 8 7 6 5

• S_[k,n] = (10-5+1)*(10+5)/2

• 2S_[k,n] = 6 * 15 / 2

• S_[k,n] = 90 / 2 = 45

For any single number missing from the sequence, by subtracting the others from the sum of 45, the remainder will be the missing number.