Math Help - Calculate some weird RV's variance

Can you help me with something? I get stuck everytime i try to solve the following problem.

Let Xi, 1<=i<=10 be independent random variables such that EXi = 10 and Var Xi = 6. Let
Calculate Var Y.

I tried to solve it using the fact that Var Y = Cov(Y,Y). That leads to:





The second term can be decomposed in two types:

1. 
2. 

The second type is zero because the variables are independent. The first type can be calculated:




Ok. But here's the problem: look what i get when I calculate Cov(XiXi+1,XiXi+1):






And I can't get rid of this E(X_{i+1)^2)!

I appreciate any help. Thanks

i haven't not checked your working at all, but:

its always true that:


so

Jeepers! How could i not realize it was that simple?

Thank you so much!