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

$\dfrac{2}{4} = \dfrac{1}{2}$

If we divide the third and second terms, we have

$\dfrac{1}{2} = \dfrac{1}{2}$

Finally, if we divide the last two terms we have

$\dfrac{\frac{1}{2}}{1} = \dfrac{1}{2}$

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