A, B, C uncorrelated random variable. X=A+B, Y=B+C. Correlation of X and Y?

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Suppose that A, B and C are uncorrelated random variables with means u1, u2, u3 and standard deviation s1, s2 and s3 respectively. If X=A+B and Y=B+C, what is the correlation of X and Y?

X=A+B, Y=B+C

Corr(X,Y) = Cov(X,Y)/(√Var(X)*√Var(Y))

Cov(X,Y) = Cov(A+B,B+C) = Var(B)+0 = (s2)²

Var(X) = Var(A+B) = s1² + s2²

Var(Y) = Var(B+C) = s2² + s3²

So : Corr(X,Y)= (s2)²/[√(s1² + s2²)√(s2² + s3²)]