gracie wrote:

x and y are the first and last terms of a sequence of consecutive positive odd integers with a range of 22. If y^2-x^2 equals the square of the arithmetic mean of the sequence, what is the sum of x and y?

A. 76

B. 80

C. 84

D. 88

E. 92

- We are given set {x,x+2,x+4,....,x+22} which \(x+22=y\) (from the question).

- The number inside the set is 12 (you can write it down one by one from x to x+22).

- So, the mean of the set must be the median, which is the average of 6th and 7th number, or \((x+10)\) and \((x+12)\).

Thus, we can calculate :

- \(y^2-x^2=(mean)^2\)

- Substitute \(y\) with \(x+22\) and mean with the median, we get : \((x+22)^2-x^2=(\frac{(x+10)+(x+12)}{2})^2\)

- Solve the equation, we get two roots of x, \(x=-11\) and \(x=33\). Since the question tells that x is a positive, so we can choose 33.

- If \(33=x\), then \(y=x+22=55\). So \(x+y=88\).

D.

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