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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

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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Find derivatives of using multiple formulae

Full solution

Q. 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

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

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