need help with the proofs

(2) (5 pts) Consider a finite geometric sequence of positive numbers with an odd number of terms. Show that the median of this geometric sequence coincides with its geometric mean. (Hint: Let the terms of the geometric sequence be given by a, ar, ar', ..., ar" , where a > 0 and r > 0 and n is an odd number.) (3) (10 pts) Let 11, 12, ..., In be any real numbers and let I = (x1 + ... + In) be the sample mean of these numbers. Recall that the sample variance of these numbers is given by $2 ( I; - I)2. n - (i) Show that shifting the observations by a constant does not change the sample variance. That is, show that for any constant c, the sample variance of I1 + c, 12 + C, ..., I, + c is the same as that of the original observations I1, 12, . .., In- (ii) Show that s' can also be calculated from the formula