Question

Determine whether the given sequence could be geometric, arithmetic, or neither. if possible, identify either the common ratio or the common difference. 4, 2, 1, 1/2

Answer 1

ARITHMETIC CHECK: A sequence is said to be arithmetic if any two consecutive terms differ by the same constant.

So, the test to check if a series is arithmetic is to compute consecutive differences, and see if they all return the same number.

If we subtract the first two terms, we have
`2-4 = -2`. If we subtract the third and second terms, we have
`1-2 = -1`.

These two differences returned two different values, so the series is not arithmetic.

GEOMETRIC CHECK: A sequence is said to be geometric if any two consecutive terms are in the same ratio.

So, the test to check if a series is geometric is to compute consecutive ratios, and see if they all return the same number.

If we divide the first two terms, we have

`(2)/(4) = (1)/(2)`

If we divide the third and second terms, we have

`(1)/(2) = (1)/(2)`

Finally, if we divide the last two terms we have

`((1)/(2))/(1) = (1)/(2)`

So, all ratios return the same number. This means that the series is geometric, and the common ratio is 1/2