Question 2 We are going to find the condition necessary for

Question 2: We are going to find the condition necessary for (X 1 ari) 2. (As we saw in class, in general 1 a? (Xi 1 ari)2. But under some very specific conditions i-1 ri) is true. d) For n 3, write explicitly X z and (X 1 ari) (3 points) e) Then what has to be true of a1, aro, T3 for Li 1 i (y i 1 zi) to hold when n 3? 2 3 points f) For any n, show that (YCI 1 z? +S 1 1 ariai. (A bit hard, use previous question to develop an intuitive understanding, you can talk your way through some steps if needed. (3 points) g) Then what has to be true of Xi 1 1iti aria for YCI 1 ar? 3(Yi 1 ari to hold? (3 points

Solution

d.

Sum( xi^2), n = 1 to 3 = x1^2 + x2^2 + x3^2 [ANSWER]

[Sum (xi), n = 1 to 3]^2 = (x1 + x2 + x3^2)^2 = x1^2 + + x2^2 + x3^2 + 2 x1 x2 + 2 x2 x3 + 2 x1 x3 [ANSWER]

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e.

As

Sum( xi^2), n = 1 to 3 = x1^2 + x2^2 + x3^2

[Sum (xi), n = 1 to 3]^2 = (x1 + x2 + x3^2)^2 = x1^2 + + x2^2 + x3^2 + 2 x1 x2 + 2 x2 x3 + 2 x1 x3   

Then

x1^2 + x2^2 + x3^2 = x1^2 + + x2^2 + x3^2 + 2 x1 x2 + 2 x2 x3 + 2 x1 x3   

0 = 2 x1 x2 + 2 x2 x3 + 2 x1 x3   

0 = x1 x2 + x2 x3 + x1 x3 [ANSWER]

The equation above must hold for the condition to be true.

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