Independent random variables X₁, X₂,.. , X_n having U([0, 1])-uniform distribution and upper record values in this set are considered. We study the problem of maximizing (taking into account some consecutively observed values x₁, x₂,.. , x_k of these X’s) the expectation of sums of records in this sequence under the optimal choice of the corresponding variable X_k (instead of X₁) as the initial record value.