Question

Consider the sequence 100, 50, 25,12.5 ... Can it arithmetic? Can it be geometric? why?

Answer 1

Answer:

(a) Arithmetic? No. Geometric? YES!

(b) The terms will never become zero or negative.

Step-by-step explanation:

Explaining part (a)

An aritmetic sequence has the same difference between each term, positive or negative.

Example:  1, 4, 7, 10, 13, ...  the difference between consecutive terms is always 3

Example:  0, -2, -4, -6, -8, ... the difference between consecutive terms is always -2

A geometric sequence has the same ratio (or multiplier) between consective terms, positive or negative, causing an increase or a decrease.

Example: 1, 2, 4, 8, 16, ...  each term is 2x the term before it

Example: 64, 32, 16, 8, 4, 2, 1, (1/2), (1/4), (1/8), ...each term is half the term before

Example: +91, -27, +9, -3, +1, -1/3, +1/9. -1/27, ... each term is -1/3 of term before

In the given sequence, we can see that the difference between terms is not always the same number. 100 - 50 = 50, but 50 - 25 = 25, not 50.

However, we do see that each term is half the term before it. That is definitely geometric.

Explaining part (b)

For 100, 50, 25, 12.5, ... we get each term by multiplying the previous term by (1/2), which is a positive number. No matter how many times you multiply by a positive, you'll never get a negative or get a zero.

Multiply by 1/2 enough times, and you get as small a positive number as you like, but it will always be positive, even if you get down to billionths or trillionths.