Question

Does the geometric sequence converge or diverge? Explain.

5,-2.5, 1.25,-0.625,...

O The sequence diverges because r = -2, which is less than 1.
The sequence converges because |r | = 0.5, which is less than 1.
O The sequence diverges because Ir | = 2, which is greater than 1.
O The sequence converges because r = 0.5, which is less than 1.

Answer 1

In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor called the common ratio (denoted by
`\displaystyle\sf r`). In the given sequence, the common ratio is
`\displaystyle\sf r=-2`.

To determine if the sequence converges or diverges, we look at the absolute value of the common ratio (
`\displaystyle\sf |r|`). If the absolute value of the common ratio is less than 1 (
`\displaystyle\sf |r|<1`), the sequence converges. If the absolute value of the common ratio is greater than or equal to 1 (
`\displaystyle\sf |r|\geq 1`), the sequence diverges.

In this case, the absolute value of the common ratio is
`\displaystyle\sf |-2|=2`, which is greater than 1. Therefore, the sequence diverges.

Answer 2

Explanation:

The common ratio in a geometric sequence is calculated by dividing any term by its preceding term. In this case:

-2.5 ÷ 5 = -0.5

1.25 ÷ -2.5 = -0.5

-0.625 ÷ 1.25 = -0.5

We can observe that the common ratio between each term is -0.5.

A geometric sequence converges if the absolute value of the common ratio is between -1 and 1. In this case, the absolute value of the common ratio (-0.5) is less than 1. Therefore, the geometric sequence converges.

In a converging geometric sequence, as more terms are added, the values approach a certain limiting value. In this case, since the common ratio is negative, the terms alternate between positive and negative values. As the sequence progresses, the absolute values of the terms decrease, approaching zero.

Hence, the geometric sequence 5, -2.5, 1.25, -0.625, ... converges.