Averages

Setting up A: The n consecutive odd numbers starting with x are: x, x+2, x+4, ..., x+2(n-1)

For an arithmetic sequence, the average equals the middle term (or average of first and last terms):

1. First term: x
2. Last term: x + 2(n-1)
3. Average A = [x + (x + 2(n-1))]/2 = (2x + 2n - 2)/2 = x + n - 1

Setting up B: The n consecutive odd numbers starting with x+4 are: x+4, x+6, x+8, ..., (x+4)+2(n-1)

Using the same logic:

1. First term: x + 4
2. Last term: x + 4 + 2(n-1)
3. Average B = [(x+4) + (x+4+2(n-1))]/2 = (2x + 8 + 2n - 2)/2 = x + n + 3

Finding B - A: B - A = (x + n + 3) - (x + n - 1) = 4

Verification (and you'll cry once you know that you could have just assumed lol): Let's check with x = 1, n = 3:

• A's series: 1, 3, 5 → Average = 3
• B's series: 5, 7, 9 → Average = 7
• B - A = 7 - 3 = 4 ✓

The answer is (a) is always 4.

This makes intuitive sense: when you shift an arithmetic sequence by a constant amount (here by 4), the average shifts by that same amount, regardless of x or n.

a