The average of 50 consecutive natural numbers is x. If the next four consecutive natural numbers are also included, what will be the new average in terms of x?

Introduction / Context:
This question tests understanding of averages for consecutive natural numbers and how the average changes when more consecutive terms are added. You are told that the average of 50 consecutive natural numbers is x and you must find the new average when the next four consecutive numbers are included. Because the numbers are consecutive, their pattern is very regular and can be handled with simple algebra.

Given Data / Assumptions:
- There are 50 consecutive natural numbers.
- The average of these 50 numbers is x.
- Four more consecutive natural numbers immediately after this block are included, making a total of 54 numbers.
- All numbers are consecutive and belong to the same arithmetic progression with common difference 1.
- We need the new average expressed in terms of x.

Concept / Approach:
The average of consecutive natural numbers is equal to the middle value of the sequence. For an even count of numbers, the average is the mean of the two central numbers. When you extend a block of consecutive numbers by adding more terms at the upper end, the new average will shift toward the newer, larger values. A direct algebraic method starts by expressing the original average and then recomputing the sum and average after adding four more numbers, and then comparing it with x.

Step-by-Step Solution:
Step 1: Let the first of the 50 consecutive natural numbers be n. Then the numbers are n, n + 1, n + 2, ..., n + 49.Step 2: The average of these 50 numbers is x, given by x = (first + last) / 2 = (n + (n + 49)) / 2.Step 3: Simplify this: x = (2n + 49) / 2.Step 4: The next four consecutive numbers are n + 50, n + 51, n + 52 and n + 53.Step 5: The sum of the original 50 numbers is 50 * x.Step 6: The sum of the four new numbers is (n + 50) + (n + 51) + (n + 52) + (n + 53) = 4n + 206.Step 7: New total sum for 54 numbers = 50x + 4n + 206.Step 8: Use x = (2n + 49) / 2 to replace 4n + 206. From x, we get 2x = 2n + 49, so 4n = 2 * (2n) = 4x - 98. Therefore 4n + 206 = 4x - 98 + 206 = 4x + 108.Step 9: New total sum = 50x + (4x + 108) = 54x + 108.Step 10: New average = (54x + 108) / 54 = x + 108 / 54 = x + 2.

Verification / Alternative check:
Take a simple numerical example. Suppose the 50 consecutive numbers are 1 to 50. Their average is (1 + 50) / 2 = 25.5, so x = 25.5. If we include the next four numbers 51, 52, 53 and 54, we now have numbers 1 to 54. The new average is (1 + 54) / 2 = 27.5. The change in average is 27.5 - 25.5 = 2, which matches x + 2. This confirms that the algebraic result is correct.

Why Other Options Are Wrong:
The average cannot increase by only 1 because four larger numbers are added to the high end of the range, shifting the average more. It also cannot increase by 4 or by x / 54, which would not match either the algebraic deduction or the numerical example. Therefore only x + 2 correctly describes the new average.

Common Pitfalls:
Students sometimes assume that the average remains the same for any extended set of consecutive numbers, which is not true when you only extend at one end. Others forget that the average is closely tied to the first and last terms in an arithmetic progression. Remember that for consecutive numbers, the average moves toward the side where you add more terms and the change can be calculated exactly using sums and counts.

Final Answer:
The new average when the next four numbers are included is x + 2.