The average of any 5 consecutive odd natural numbers is k. If two more consecutive odd natural numbers, immediately following those 5 numbers, are added to the set, what will be the new average in terms of k?

Introduction / Context:
This problem involves averages of consecutive odd natural numbers and how the average changes when more terms are added. The question is algebraic in nature and is intended to test your ability to represent patterns using variables, compute sums of sequences and manipulate averages symbolically. Recognising the symmetry of consecutive odd numbers around their middle term is the key insight.

Given Data / Assumptions:

• We have 5 consecutive odd natural numbers.
• The average (mean) of these 5 numbers is k.
• Two more consecutive odd numbers, immediately after these 5, are added to the set.
• We must find the new average of all 7 numbers in terms of k.
• All numbers are positive odd integers.

Concept / Approach:
Five consecutive odd numbers can be written in terms of their middle number. If the middle number is n, then the five numbers are n - 4, n - 2, n, n + 2 and n + 4. The average of these five numbers is exactly the middle one, n. We are told that this average is k, so n = k. When we add the next two consecutive odds, n + 6 and n + 8, we can compute the new sum and average using algebra.

Step-by-Step Solution:
Step 1: Let the five consecutive odd numbers be n - 4, n - 2, n, n + 2 and n + 4. Step 2: Their average is the middle term n, and we are given that this average is k, so n = k. Step 3: The next two consecutive odd numbers after n + 4 are n + 6 and n + 8. Step 4: Now the full set of 7 numbers is: n - 4, n - 2, n, n + 2, n + 4, n + 6, n + 8. Step 5: Sum of these 7 numbers = 7n + ( -4 - 2 + 0 + 2 + 4 + 6 + 8 ). Step 6: Compute the constant part: -4 - 2 + 0 + 2 + 4 + 6 + 8 = 14, so sum = 7n + 14. Step 7: New average = (7n + 14) / 7 = n + 2. Step 8: Replace n with k, since n = k, to get new average = k + 2.

Verification / Alternative check:
Use a concrete example to confirm the algebra. Suppose the 5 consecutive odd numbers are 3, 5, 7, 9 and 11. Their average is (3 + 5 + 7 + 9 + 11) / 5 = 35 / 5 = 7, so k = 7. The next two odd numbers are 13 and 15. Now the 7 numbers are 3, 5, 7, 9, 11, 13 and 15. The new average is (3 + 5 + 7 + 9 + 11 + 13 + 15) / 7 = 63 / 7 = 9. This equals k + 2 = 7 + 2, which verifies our general result.

Why Other Options Are Wrong:
Options such as 2(k + 1), 2k - 3 and 2k + 1 incorrectly assume that the average doubles or follows some other incorrect pattern and do not match the actual algebraic derivation. For instance, 2(k + 1) would give 16 for k = 7, which is much larger than the correct new average 9 in our example. The option k - 2 suggests a decrease in the average, which is impossible because we are adding larger numbers to the original set.

Common Pitfalls:
A common mistake is to think that the new average is simply k plus the average of the two added numbers divided by some factor, without recalculating the total sum. Others may misrepresent the five consecutive odd numbers or incorrectly compute the constant part of the sum. Some students also forget that adding larger numbers to a set will generally increase the average. Writing the sequence clearly and using algebraic expressions prevents these errors.

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