Total sum of the 8 numbers = 8 * 14 = 112. Total sum of the 6 numbers = 6 * 16 = 96. Sum of the remaining 2 numbers = 112 - 96 = 16. The average of these 2 remaining numbers is 16 / 2 = 8.

Similar to addition, when a constant value is subtracted from every number in a dataset, the average decreases by that exact constant. Thus, the new average is 50 - 10 = 40.

When a constant value is added to every observation in a dataset, the average of the dataset increases by that same constant. Therefore, the new average will be 25 + 5 = 30.

If every number in a dataset is multiplied by a constant factor 'k', the new average also becomes 'k' times the original average. Here, the original average is 40 and k = 3. The new average is 40 * 3 = 120.

Let the even numbers be x, x+2, x+4, and x+6. Their sum is 4x + 12. The average is (4x + 12) / 4 = x + 3. Given x + 3 = 27, we find x = 24. The largest number is x + 6 = 24 + 6 = 30.

Since there are 7 numbers (an odd count), the average, 25, is the middle number. The sequence is: _, _, _, 25, _, _, _. The three odd numbers before 25 are 23, 21, and 19. Thus, the smallest number is 19.

For an odd number of consecutive integers, the average is always the middle number. So, the middle number is 20. The numbers are 18, 19, 20, 21, and 22. The largest number is 22.

The average of the first 'n' odd natural numbers is simply 'n'. Therefore, for the first 10 odd numbers, the average is 10. The sum of the first 10 odd numbers is 10^2 = 100, and 100 / 10 = 10.

The formula for the average of the first 'n' even natural numbers is (n + 1). For the first 10 even numbers, the average is 10 + 1 = 11. Alternatively, the numbers are 2, 4, ..., 20. Sum = 110. Average = 110 / 10 = 11.

The sum of the first 'n' natural numbers is given by the formula n(n+1)/2. For n=10, the sum is 10(11)/2 = 55. The average is the sum divided by n, which is 55 / 10 = 5.5.