answer:The harmonic mean is calculated as the reciprocal of the arithmetic mean of the reciprocals of the given numbers. Let's denote the numbers as a1, a2, and a3, where a1 = 15, a2 = 13/3, and a3 = 8/3. The harmonic mean (H) is given by: [ H = frac{3}{frac{1}{a1} + frac{1}{a2} + frac{1}{a3}} ] Calculating the reciprocals: [ frac{1}{a1} = frac{1}{15}, frac{1}{a2} = frac{3}{13}, frac{1}{a3} = frac{3}{8} ] Now, the arithmetic mean of these reciprocals: [ frac{1}{3} left( frac{1}{15} + frac{3}{13} + frac{3}{8} right) ] Combining the fractions: [ frac{1}{3} left( frac{13 cdot 8 + 3 cdot 15 cdot 8 + 15 cdot 13 cdot 3}{15 cdot 13 cdot 8} right) ] [ frac{1}{3} left( frac{104 + 360 + 585}{15 cdot 13 cdot 8} right) ] [ frac{1}{3} left( frac{1049}{15 cdot 13 cdot 8} right) ] Now, calculating the harmonic mean: [ H = frac{3}{frac{1049}{15 cdot 13 cdot 8}} ] [ H = frac{3 cdot 15 cdot 13 cdot 8}{1049} ] [ H = frac{4680}{1049} ] So, the harmonic mean is frac{4680}{1049}.

answer:To determine the number of parallelograms formed by N lines, we can use the following steps: 1. Group the lines into parallel classes, where lines in the same class are parallel to each other but not to lines in other classes. Let's denote the number of parallel classes as k and the number of lines in each class as a1, a2, ..., ak. 2. The number of parallelograms formed by lines in two parallel classes Ai and Aj is given by the formula: Number of parallelograms = (a_i * (a_i - 1)) / 2 * (a_j * (a_j - 1)) / 2 3. Sum up the number of parallelograms formed by all pairs of parallel classes: Total number of parallelograms = ∑∑ (a_i * (a_i - 1)) / 2 * (a_j * (a_j - 1)) / 2 for 1 ≤ i < j ≤ k In the example you provided, we have k = 2, a_1 = 2, a_2 = 2, and a_3 = 1. Using the formula, we can calculate the number of parallelograms as: Number of parallelograms = (2 * (2 - 1)) / 2 * (2 * (2 - 1)) / 2 = 1 Therefore, there is 1 parallelogram formed in this example.

answer:The claim that if a formula Q is not a logical consequence of a set of formulas C, then its negation neg Q is a logical consequence of C, is not generally true. To see why, let's examine the definitions involved. Logical consequence (from the definition provided) means that if every interpretation that makes all formulas in C true also makes Q true, then Q is a logical consequence of C. Conversely, if Q is not a logical consequence of C, there exists at least one interpretation that makes all formulas in C true while making Q false. Now, for neg Q to be a logical consequence of C, every interpretation that makes all formulas in C true should also make neg Q (the negation of Q) true, which means Q would be false. However, this contradicts the previous scenario where there is at least one interpretation that makes Q false while preserving the truth of formulas in C. To provide a counterexample: Consider the set of formulas C = { p lor q }, and let Q = p. It is clear that p lor q does not logically entail p, as there is a truth assignment where v(p) = text{false} and v(q) = text{true}, which makes p lor q true but p false. Furthermore, p lor q also does not entail neg p, because there is a truth assignment where v(p) = text{true}, which makes p lor q and neg p both false. Hence, we have a counterexample demonstrating that the negation of a formula that is not a logical consequence of a set of formulas is not necessarily a logical consequence of that set.

answer:To calculate the present value of the annuity, we can break it down into three components: the level payment, the increasing payments, and the decreasing payments. 1. **Level Payment:** - The level payment is the constant amount paid before the payments start to increase. - In this case, the level payment is 900 (1000 - 100). - The present value of the level payment can be calculated using the formula for the present value of an annuity-immediate: PV_{level} = A cdot frac{1 - v^n}{i} where A is the level payment, v is the discount factor, n is the number of payments, and i is the effective annual rate of interest. 2. **Increasing Payments:** - The increasing payments are the payments that increase by 100 each year until they reach 2000. - The present value of the increasing payments can be calculated using the formula for the present value of an increasing annuity: PV_{increasing} = A cdot v cdot frac{1 - v^n}{i - g} where A is the initial payment, v is the discount factor, n is the number of payments, i is the effective annual rate of interest, and g is the annual increase in payments. 3. **Decreasing Payments:** - The decreasing payments are the payments that decrease by 200 each year after reaching 2000. - The present value of the decreasing payments can be calculated using the formula for the present value of a decreasing annuity: PV_{decreasing} = A cdot v^n cdot frac{1 - v^m}{i + g} where A is the final payment, v is the discount factor, n is the number of payments, i is the effective annual rate of interest, and g is the annual decrease in payments. 4. **Total Present Value:** - The total present value of the annuity is the sum of the present values of the level payment, increasing payments, and decreasing payments: PV_{total} = PV_{level} + PV_{increasing} + PV_{decreasing} By plugging in the given values and using the appropriate formulas, we can calculate the present value of the annuity.