Points to Remember:
- Consecutive odd numbers have a constant difference of 2.
- The average of a series of consecutive numbers is the middle number (or the average of the two middle numbers if there’s an even number of terms).
- Arithmetic progression concepts are relevant.
Introduction:
This question requires a factual and analytical approach to determine the average of a sequence of consecutive odd numbers. The problem involves an arithmetic progression where the common difference is 2. We are given that the sequence starts with an odd number ‘a’. Understanding arithmetic progressions and their properties will allow us to efficiently solve this problem without needing to explicitly calculate each number in the sequence.
Body:
1. Defining the Sequence:
The six consecutive odd numbers starting from ‘a’ can be represented as: a, a+2, a+4, a+6, a+8, a+10.
2. Calculating the Sum:
The sum of an arithmetic series can be calculated using the formula: Sum = (n/2) * [2a + (n-1)d], where ‘n’ is the number of terms, ‘a’ is the first term, and ‘d’ is the common difference. In our case, n=6, a=a, and d=2.
Therefore, the sum of our sequence is: Sum = (6/2) * [2a + (6-1)2] = 3 * [2a + 10] = 6a + 30.
3. Calculating the Average:
The average is calculated by dividing the sum by the number of terms: Average = Sum / n = (6a + 30) / 6 = a + 5.
4. Alternative Approach (Intuitive):
Since the numbers are consecutive and evenly spaced, the average will always be the middle value. In a sequence of six numbers, the middle values are the 3rd and 4th numbers. Their average is (a+4 + a+6)/2 = (2a + 10)/2 = a+5. This confirms our previous calculation.
Conclusion:
The average of six consecutive odd numbers starting from ‘a’ is always ‘a + 5’. This result is independent of the starting odd number ‘a’. This is a direct consequence of the properties of arithmetic progressions and the symmetrical distribution of values around the mean. This simple formula provides a quick and efficient method for calculating the average of any such sequence. No further policy recommendations or best practices are needed as this is a purely mathematical problem. The solution highlights the elegance and efficiency of mathematical formulas in solving seemingly complex problems.