Which of the following expressions are equal to 30,600 ? I. quadsum_(n=1)^(45)30 n-10 II. quadsum_(n=5)^(49)20+30(n-5) III. quadsum_(n=2)^(46)-10+30(n-2) A. I only B. I and II C. II and III D. I, II, and III

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Evaluate Expression I:  Evaluate Expression I: $ \sum_{n=1}^{45} (30n - 10) $ Calculation: Each term in the sum is $30n - 10$. Summing from $n=1$ to $n=45$, $$ \sum_{n=1}^{45} (30n - 10) = 30\sum_{n=1}^{45} n - 10 \times 45 $$ $$ \sum_{n=1}^{45} n = \frac{45 \times (45 + 1)}{2} = 1035 $$ $$ 30 \times 1035 - 450 = 31050 - 450 = 30600 $$Evaluate Expression II:  Evaluate Expression II: $ \sum_{n=5}^{49} (20 + 30(n-5)) $ Calculation: Simplify the expression inside the sum, $$ 20 + 30(n-5) = 30n - 150 + 20 = 30n - 130 $$ Summing from $n=5$ to $n=49$, $$ \sum_{n=5}^{49} (30n - 130) = 30\sum_{n=5}^{49} n - 130 \times (49 - 5 + 1) $$ $$ \sum_{n=5}^{49} n = \frac{49 \times (49 + 1)}{2} - \frac{4 \times (4 + 1)}{2} = 1225 - 10 = 1215 $$ $$ 30 \times 1215 - 130 \times 45 = 36450 - 5850 = 30600 $$Evaluate Expression III:  Evaluate Expression III: $ \sum_{n=2}^{46} (-10 + 30(n-2)) $ Calculation: Simplify the expression inside the sum, $$ -10 + 30(n-2) = 30n - 60 - 10 = 30n - 70 $$ Summing from $n=2$ to $n=46$, $$ \sum_{n=2}^{46} (30n - 70) = 30\sum_{n=2}^{46} n - 70 \times (46 - 2 + 1) $$ $$ \sum_{n=2}^{46} n = \frac{46 \times (46 + 1)}{2} - 1 = 1035 - 1 = 1034 $$ $$ 30 \times 1034 - 70 \times 45 = 31020 - 3150 = 27870 $$

Which of the following expressions are equal to $30,600$ ? $\newline$ I. $\sum_{n=1}^{45} 30n-10$ $\newline$ II. $\sum_{n=5}^{49} 20+30(n-5)$ $\newline$ III. $\sum_{n=2}^{46} -10+30(n-2)$ $\newline$ A. I only $\newline$ B. I and II $\newline$ C. II and III $\newline$ D. I, II, and III

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