int[] arr1 = {1, 2, 3, 4, 5}; // {1, 2, 3, 4, 5}
int[] arr2 = {1, 2}; // {1, 2}
int[] sumArr12 = arr1.Select((value, index) => value + arr2[index % arr2.Length]).ToArray(); // {2, 4, 4, 6, 6}

int[] arrOne = { 1, 2, 3, 4, 5 };
int[] arrTwo = { 1, 2 };

int[] result = new int [arrOne.Length > arrTwo.Length ? arrOne.Length : arrTwo.Length];
var firstIndex = 0;
var secondIndex = 0;

for(int index = 0; index < result.Length; index++)
{
    firstIndex = arrOne.Length > firstIndex ? firstIndex : 0;
    secondIndex = arrTwo.Length > secondIndex ? secondIndex : 0;
    result[index] = arrOne[firstIndex++] + arrTwo[secondIndex++];
}

//result = {2, 4, 4, 6, 6}

int[] arrOne = { 1, 2, 3 };
int[] arrTwo = { 1, 2, 3, 4, 5, 6 };

static IEnumerable<T> Looped<T>(T[] array)
{
    for (int i = 0; ; i = (i + 1) % array.Length)
        yield return array[i];
}

int[] ar1 = { 1, 2, 3, 4, 5 };
int[] ar2 = { 1, 2 };
var ar3 = ar1.Zip(Looped(ar2), (x, y) => x + y).ToArray();