answer:1. **Count the Total Number of Line Segments:** First, we note that there are (10) rows and (10) columns in the square array. Since each row and each column have (9) line segments between the dots, the total number of potential line segments is: [ 9 text{ line segments per row} times 10 text{ rows} + 9 text{ line segments per column} times 10 text{ columns} = 90 + 90 = 180 ] 2. **Define Variables for Line Segments:** Let ( B ) be the number of blue segments and ( R ) be the number of red segments. As given in the problem, there are (98) green line segments. Hence, we have: [ B + R + 98 = 180 ] Solving for ( B + R ): [ B + R = 180 - 98 = 82 ] 3. **Count the Connections Involving Red Dots:** Given there are (52) red dots, including (2) at the corners, (16) on the edges (excluding corners), and the rest (i.e., (52 - 2 - 16 = 34)) inside the array. - **Corner dots:** Each has (2) segments connecting them. - **Edge dots (excluding corners):** Each has (3) segments connecting them. - **Interior dots:** Each has (4) segments connecting them. Thus, the total number of line segments coming from the red dots is: [ 2 times 2 + 16 times 3 + 34 times 4 = 4 + 48 + 136 = 188 ] 4. **Distribute the Segments from Red Dots:** Out of the total (188) segments coming from red dots, (98) are green segments (since each green segment connects a red dot to a blue dot exactly once). The remaining segments are red segments connecting two red dots, and there are: [ 188 - 98 = 90 ] Since each red segment has two red dots at both ends, each red segment is counted twice. Therefore, the actual number of red segments (R) is: [ frac{90}{2} = 45 ] 5. **Calculate the Number of Blue Segments:** Now, using the relation ( B + R = 82 ): [ B + 45 = 82 implies B = 82 - 45 = 37 ] # Conclusion: Therefore, the number of blue line segments is: [ boxed{text{(B) 37}} ]

answer:1. **Scale Conversion:** The scale is 1" = 250 miles. To find the real distance in miles corresponding to inches on the map, multiply the inches by the scale: [ d_1 = 6" times 250 text{ miles/inch} = 1500 text{ miles} ] [ d_2 = 12" times 250 text{ miles/inch} = 3000 text{ miles} ] 2. **Area Calculation:** The area (A) of a rhombus can be calculated using the formula involving its diagonals: [ A = frac{1}{2} times d_1 times d_2 ] Substitute the values: [ A = frac{1}{2} times 1500 times 3000 = frac{1}{2} times 4500000 = 2250000 text{ square miles} ] Therefore, the area of the rhombus-shaped plot of land is 2250000 text{ square miles}. The final answer is The final correct answer given the choices is boxed{B} - 2250000 square miles.

answer:1. First, identify the approximate values of ( S ) and ( T ) on the number line: - ( S ) is approximately 1.5 - ( T ) is approximately 1.6 2. Next, compute ( frac{S}{T} ): [ frac{S}{T} approx frac{1.5}{1.6} ] To simplify ( frac{1.5}{1.6} ), perform the division: [ frac{1.5}{1.6} = frac{1.5 times 10}{1.6 times 10} = frac{15}{16} ] 3. Now calculate ( frac{15}{16} ): [ frac{15}{16} = 0.9375 ] 4. Determine which point on the number line corresponds to 0.9375: - Observe that point ( R ) is slightly less than 1, whereas points ( P ), ( Q ), and ( U ) represent values greater than or equal to 1. Therefore, ( R ) is the only option that closely matches 0.9375. Conclusion: [ boxed{C} ]

answer:(1) From the given conditions, we can set up the following system of equations: begin{cases} b_2 + a_1 + a_2 = 8 a_1 + a_2 = (b_2 + 1)q end{cases} Given that a_1 = 2, we have: begin{cases} q + 2 + a_2 = 8 2 + a_2 = (q + 1)q end{cases} Eliminating a_2, we get the quadratic equation: q^2 + 2q - 6 = 0 The solutions to this equation are q = 2 or q = -4. However, since q must be positive (all terms of {b_n} are positive), we discard q=-4. Thus, we have q=2, and a_2 can be calculated: a_2 = (q + 1)q - 2 = (2 + 1) cdot 2 - 2 = 4 Since a_1 = 2, we now have two terms of the arithmetic sequence, which is enough to determine the common difference d: d = a_2 - a_1 = 4 - 2 = 2 So, the general term of the arithmetic sequence is: a_n = a_1 + (n - 1)d = 2 + (n - 1) cdot 2 = 2n For the geometric sequence, since b_1 = 1 and q = 2, the general term is: b_n = b_1 cdot q^{n-1} = 1 cdot 2^{n-1} = 2^{n-1} (2) Using the general terms obtained in (1), we express c_n as: c_n = frac {a_{n}}{b_{n}} = frac {2n}{2^{n-1}} = 2n cdot left(frac {1}{2}right)^{n-1} We want to find the sum of the first n terms of the sequence {c_n}, T_n: T_n = c_1 + c_2 + ldots + c_n = 2 cdot left(frac {1}{2}right)^0 + 4 cdot left(frac {1}{2}right)^1 + ldots + 2n cdot left(frac {1}{2}right)^{n-1} Multiplying T_n by frac {1}{2}: frac {1}{2}T_n = 2 cdot left(frac {1}{2}right)^1 + 4 cdot left(frac {1}{2}right)^2 + ldots + 2n cdot left(frac {1}{2}right)^n Subtracting the second equation from the first, we get: frac {1}{2}T_n = 2 cdot left(frac {1}{2}right)^0 + 2 cdot left(frac {1}{2}right)^1 + ldots + 2 cdot left(frac {1}{2}right)^{n-1} - 2n cdot left(frac {1}{2}right)^{n} This resembles the sum of a geometric sequence, which we can calculate as follows: frac {1}{2}T_n = frac {2[1-( frac {1}{2})^{n}]}{1- frac {1}{2}} - 2n cdot left(frac {1}{2}right)^{n} = 4 - (4 + 2n) cdot left(frac {1}{2}right)^{n} Thus, the sum of the first n terms of {c_n} is: T_n = 2 cdot frac {1}{2}T_n = 8 - (4 + 2n) cdot left(frac {1}{2}right)^{n} = boxed{8 - (4 + 2n) cdot left(frac {1}{2}right)^{n}}