The sum of nine consecutive odd number of set A is 657. What is the sum of…

CONCEPT: For n consecutive odd numbers starting from a first term a, the numbers are a, a+2, a+4, ..., a+2(n-1). Their sum equals na + 2(0+1+2+...+(n-1)) = na + n(n-1). This identity lets you find either the first term (given the sum) or the sum (given the first term) for any run of consecutive odd numbers.

1. Let the lowest odd number of Set A be x. Nine consecutive odd numbers starting at x sum to 9x + 9(8) = 9x + 72.

2. Given this sum is 657: 9x + 72 = 657, so 9x = 585, so x = 65.

3. The lowest number of the second set is 18 more than Set A's lowest number: 65 + 18 = 83.

4. Apply the same identity for seven consecutive odd numbers starting at 83: sum = 7(83) + 7(6) = 581 + 42 = 623.

CROSS-CHECK: Directly adding the seven terms 83, 85, 87, 89, 91, 93, 95 also gives 623, confirming the result from the identity.