Question

Is this sequence arithmetic or geometric?

0.5, 0.75, 0.875, 0.9375, 0.96875, 0.984375

Answer 1

Neither, but is described by the formula:

`a_n = 1-2^(-n)`

Explanation:

Given:

`0.5, 0.75, 0.875, 0.9375, 0.96875, 0.984375`

This sequence has no common ratio, as we can see from:

`0.75/0.5 = 1.5 != 1.1bar(6) = 0.875/0.75`

So it is not a geometric sequence.

Looking at the differences between successive terms, we find:

`0.25, 0.125, 0.0625, 0.03125, 0.015625`

This sequence is non-constant, so the given sequence has no common difference and is not an arithmetic sequence.

However, note that this sequence of differences is itself a geometric sequence, with common ratio `0.5` in that:

`0.125/0.5 = 0.0625/0.125 = 0.03125/0.0625 = 0.0156625/0.03125 = 0.5`

If we express the given sequence as fractions, then we get:

`1/2, 3/4, 7/8, 15/16, 31/32`

from which we can recognise a pattern and write a formula:

`a_n = (2^n-1)/2^n = 2^n/2^n-1/2^n = 1-2^(-n)`