Show work and explain with formulas.

23. find the sum of the geometric series: 1/9 + 1/3 + 1 + + 2187

24. in a geometric sequence, the first term is 27 and the common ratio is 1/3. find the sum of the first 6 terms.

25. the sum of the first n terms of the geometric sequence is -6, -12, -24, is -762. find the value of n.

Acircular hot tub located near the middle of a rectangular patio has a diameter of 48 inches. the patio measures 11 feet by 6 feet. what is the area of the patio that is not covered by the hot tub?

Consider the following estimates from the early 2010s of shares of income to each group. country poorest 40% next 30% richest 30% bolivia 10 25 65 chile 10 20 70 uruguay 20 30 50 1.) using the 4-point curved line drawing tool, plot the lorenz curve for bolivia. properly label your curve. 2.) using the 4-point curved line drawing tool, plot the lorenz curve for uruguay. properly label your curve.

Given the expression (7 ? 4i) ? (2 + 6i), perform the indicated operation and write the answer in the form a + bi.