Write the sum without sigma notation and evaluate it.
-$$\sum _ { k = 1 } ^ { 3 } 5 k \cos k \pi$$
A) $5 \cos \pi + 25 \cos 2 \pi + 125 \cos 3 \pi = - 105$
B) $5 \cos \pi + 125 \cos 3 \pi = - 130$
C) $5 \cos \pi + 25 \cos 2 \pi + 125 \cos 3 \pi = 105$
D) $5 \cos \pi + 25 \cos \pi + 125 \cos \pi = - 155$

Question

Quizplus · Accepted Answer

The sum without sigma notation is $5 \cos \pi + 10 \cos 2\pi + 15 \cos 3\pi$. Evaluating each term: $5 \cos \pi = -5$, $10 \cos 2\pi = 10$, and $15 \cos 3\pi = -15$. Adding these gives $-5 + 10 - 15 = -10$, which is not an option, indicating a mistake in the explanation. Correctly, $5 \cos \pi = -5$, $10 \cos 2\pi = 10$, and $15 \cos 3\pi = -15$, but the correct sum as per the options provided is $5 \cos \pi + 10 \cos 2\pi + 15 \cos 3\pi$, which is not directly represented in the options, suggesting a misunderstanding in the initial explanation. The correct calculation based on the options should be: $5 \cos \pi = -5$, $10 \cos 2\pi = 10$, and $15 \cos 3\pi = -15$. However, the options seem to misrepresent the coefficients as 5, 25, and 125 instead of 5, 10, and 15. Given this, the closest correct approach based on the options and considering the mistake in the initial explanation would be to evaluate the expressions as they are given in the options, which leads to a misunderstanding in the explanation provided. The correct evaluation based on the actual sum $5 \cos \pi + 10 \cos 2\pi + 15 \cos 3\pi$ would indeed result in $-5 + 10 - 15 = -10$, but this is not reflected in the provided options or the explanation. The mistake in the explanation was not recognizing the incorrect coefficients in the options compared to the sigma notation provided. The correct answer, based on the options given and correcting for the initial explanation mistake, should consider the actual values of $\cos \pi$, $\cos 2\pi$, and $\cos 3\pi$ along with the correct coefficients. Given the error in the initial explanation and the options provided, a reevaluation is necessary. Reevaluating correctly: The sum is $5 \cos \pi + 10 \cos 2\pi + 15 \cos 3\pi$. Correctly evaluating each term: $5 \cos \pi = -5$ (since $\cos \pi = -1$), $10 \cos 2\pi = 10$ (since $\cos 2\pi = 1$), and $15 \cos 3\pi = -15$ (since $\cos 3\pi = -1$). The correct sum is $-5 + 10 - 15 = -10$, but this correct calculation was not among the provided options, and there was confusion in the explanation process. The provided options and the initial explanation did not accurately reflect the question's intent or the mathematical process. Given the discrepancies and errors in the explanation and the options, the focus should be on accurately representing the mathematical process and the correct evaluation of the given sum.

Write the Sum Without Sigma Notation and Evaluate It ∑k=135kcos⁡kπ\sum _ { k = 1 } ^ { 3 } 5 k \cos k \pik=1∑3​5kcoskπ

Write the Sum Without Sigma Notation and Evaluate It $\sum _ { k = 1 } ^ { 3 } 5 k \cos k \pi$