Final answer:
The common ratio r for the geometric sequence { 1, -4, 16, -64, 256, … } is found by dividing any term by the term preceding it, resulting in a common ratio of -4. The common ratio for the given geometric sequence is -4.
Step-by-step explanation:
To find the common ratio r for the following geometric sequence { 1 , − 4 , 16 , − 64 , 256 , … }, we need to divide any term in the sequence by the term preceding it.
Noticing that each term is the result of the previous term multiplied by a certain constant number, we can take for example the second term (−4) and divide it by the first term (1), which gives us −4 / 1 = −4.
As a geometric sequence has the same ratio between consecutive terms, dividing the third term (16) by the second term (−4) will also give us 16 / (−4) = −4. It can be seen that this ratio is consistent for subsequent terms as well. Therefore, the common ratio r for this sequence is −4.