To find the value of H, we can use the formula for the sum of an arithmetic series:

H = (n/2)(first term + last term)

In this case, the first term is 1 and the last term is 50. So we have:

H = (50/2)(1 + 50)
H = 25(51)
H = 1275

To find the value of K, we need to sum the products of odd numbers from 1 to 99 with their corresponding even numbers from 99 to 51. We can rewrite K as follows:

K = 1x99 + 3x97 + 5x95 + ... + 49x51

Notice that each term in K is the product of an odd number and an even number, and the difference between consecutive odd numbers is 2. Similarly, the difference between consecutive even numbers is also 2.

We can rewrite K as the sum of two arithmetic series:

K = (1x99 + 3x97 + 5x95 + ... + 49x51) + (2x98 + 4x96 + 6x94 + ... + 50x52)

Using the formula for the sum of an arithmetic series, we can calculate each part separately.

For the first part:

K1 = (n/2)(first term + last term)
= (25/2)(1 + 49)
= 25(50)
= 1250

For the second part:

K2 = (n/2)(first term + last term)
= (25/2)(2 + 50)
= 25(52)
= 1300

Finally, we can find K by adding K1 and K2:

K = K1 + K2
= 1250 + 1300
= 2550

Therefore, the value of H is 1275 and the value of K is 2550.