Write the sum without sigma notation and evaluate it.
-$$\sum _ { k = 1 } ^ { 3 } ( - 1 ) ^ { k + 1 } \cos 10 \mathrm { k } \pi$$
A) $\cos 10 \pi + \cos 30 \pi = 2$
B) $\cos 10 \pi - \cos 20 \pi + \cos 30 \pi = 1$
C) $- \cos 10 \pi + \cos 20 \pi - \cos 30 \pi = - 1$
D) $\cos 10 \pi - \cos 20 \pi + \cos 30 \pi = - 1$

Question

Quizplus · Accepted Answer

We plug in k = 1, 2, and 3 into the given expression and add up the terms:
\begin{align*}
\sum_{k=1}^3 (-1)^{k+1} \cos 10k\pi &= (-1)^2 \cos 10\pi + (-1)^3 \cos 20\pi + (-1)^4 \cos 30\pi \
&= \cos 10\pi - \cos 20\pi + \cos 30\pi \
&= 1\cdot 1 - (-1)\cdot 0 + \frac{1}{2}\cdot (-1) \
&= 1
\end{align*}
Therefore, the correct answer is B, which matches our result.

Write the Sum Without Sigma Notation and Evaluate It ∑k=13(−1)k+1cos⁡10kπ\sum _ { k = 1 } ^ { 3 } ( - 1 ) ^ { k + 1 } \cos 10 \mathrm { k } \pik=1∑3​(−1)k+1cos10kπ

Write the Sum Without Sigma Notation and Evaluate It $\sum _ { k = 1 } ^ { 3 } ( - 1 ) ^ { k + 1 } \cos 10 \mathrm { k } \pi$