The average of any five consecutive odd natural numbers is k. If two more consecutive odd natural numbers, immediately after these five numbers, are added to the set, what will be the new average in terms of k?

Introduction:
This problem examines the properties of consecutive odd natural numbers and how their average behaves when more numbers from the same sequence are added. Consecutive odd numbers form an arithmetic progression with a common difference of 2, and their average has a simple relationship with the middle term of the sequence.

Given Data / Assumptions:
- We have five consecutive odd natural numbers. - The average of these five numbers is k. - Two more consecutive odd numbers, immediately following the first five, are added. - We must find the new average in terms of k.

Concept / Approach:
For any arithmetic progression with an odd number of terms, the average is equal to the middle term. Five consecutive odd numbers can be written symmetrically around their middle value. When two more consecutive odd numbers are added after the first five, the new set has seven consecutive odd numbers. For seven consecutive odd numbers, the average is still the middle term, which can be related back to k.

Step-by-Step Solution:
Step 1: Let the five consecutive odd numbers be n - 4, n - 2, n, n + 2 and n + 4, where n is the middle odd number. Step 2: For five consecutive odd numbers, the average is the middle term, so the average is n. Step 3: The question states that this average is k, so n = k. Step 4: The next two consecutive odd numbers after n + 4 are n + 6 and n + 8. Step 5: After adding these two, we have seven numbers: n - 4, n - 2, n, n + 2, n + 4, n + 6 and n + 8. Step 6: These seven numbers are still in arithmetic progression with a common difference of 2. Step 7: For seven consecutive odd numbers, the average is again the middle term, which is the fourth number in order. Step 8: The fourth number is n + 2, since the sequence is symmetric around this value. Step 9: Substitute n = k to get the new average: k + 2.

Verification / Alternative Check:
Take a concrete example. Suppose the five consecutive odd numbers are 3, 5, 7, 9 and 11. Their average is 7, so k = 7. The next two odd numbers are 13 and 15. The seven numbers are 3, 5, 7, 9, 11, 13 and 15. The average of these seven numbers is the middle value 9. Since k = 7, k + 2 = 9, which matches the new average, confirming the formula.

Why Other Options Are Wrong:
Option k - 2 and k + 1 underestimate the shift in the average when larger numbers are added. Options 2k + 1 and 2k - 3 are not consistent dimensionally; they grow too fast compared to the linear progression of the numbers.

Common Pitfalls:
Students may incorrectly try to recompute the average using long algebraic expressions for all seven terms instead of using the symmetry of consecutive odd numbers. Remembering that for an odd count of terms in an arithmetic progression the average is equal to the middle term simplifies the problem significantly.

Final Answer:
The new average after adding the next two odd numbers is k + 2.