Provide an appropriate response.
-Which of the following express $- 3 + 9 - 27 + 81 - 243$ in sigma notation?
I. $\sum _ { k = 1 } ^ { 5 } ( - 1 ) ^ { k _ { 3 } k }$
II. $\sum _ { \mathrm { k } = 2 } ^ { 6 } ( - 3 ) ^ { \mathrm { k } - 1 }$
III. $\sum _ { \mathrm { k } = 1 } ^ { 5 } ( - 1 ) ^ { \mathrm { k } - 1 } { } _ { 3 } \mathrm { k }$
A) II only
B) I, II, and III
C) III only
D) I and II

Question

Quizplus · Accepted Answer

We can rewrite the expression as:
$$(-3)^1 + (-3)^2 + (-3)^3 + (-3)^4 + (-3)^5$$

This means that we want to sum the terms of the form $(-3)^{k-1}$ as $k$ goes from $2$ to $6$ (or equivalently, from $1$ to $5$). This is exactly what option II is doing. Option I does not include the correct terms, while option III has the correct terms but is multiplying them by an incorrect factor. Therefore, the correct choice(s) is (are) D.

Provide an Appropriate Response −3+9−27+81−243- 3 + 9 - 27 + 81 - 243−3+9−27+81−243

Provide an Appropriate Response $- 3 + 9 - 27 + 81 - 243$