The question asks which statement is true for a geometric progression with a common ratio $r = -1$. The options are: a. The terms decrease exponentially b. The terms are all zeros c. The terms alternate between positive and negative d. The terms increase exponentially e. The terms remain constant

1. Problem Description

The question asks which statement is true for a geometric progression with a common ratio $r = -1$. The options are:

a. The terms decrease exponentially

b. The terms are all zeros

c. The terms alternate between positive and negative

d. The terms increase exponentially

e. The terms remain constant

2. Solution Steps

A geometric progression is a sequence of numbers where each term is found by multiplying the previous term by a constant called the common ratio. Let the first term be $a$. Then the sequence is:

$a, ar, ar^2, ar^3, ...$

Given $r = -1$, the geometric progression becomes:

$a, -a, a, -a, a, -a, ...$

Let's analyze each option:

a. The terms decrease exponentially: This is false. The terms alternate between $a$ and $-a$. They don't decrease.

b. The terms are all zeros: This is false, unless $a = 0$. However, the question implies that the geometric progression is non-trivial (not all terms are zero).

c. The terms alternate between positive and negative: This is true as long as $a$ is not zero. If $a>0$, the terms are positive, negative, positive, negative... If $a<0$, the terms are negative, positive, negative, positive...

d. The terms increase exponentially: This is false. The terms alternate between $a$ and $-a$. They don't increase.

e. The terms remain constant: This is false, unless $a = 0$. However, the question implies that the geometric progression is non-trivial (not all terms are zero).

Therefore, the correct option is c.

3. Final Answer

c. The terms alternate between positive and negative