Can the numbers 1, 1/2, and 1/3 belong to the same arithmetic sequence?
An arithmetic sequence is a sequence of numbers in which the difference between successive numbers in that sequence is constant. The 3 you have do not conform to this, the differences are 1/2 and 1/6 (very near to a sixth) – the difference is not constant. This is an example of an arithmetic sequence… 5, 10, 15, 20, 25 – because the difference between successive numbers is constant (5). Adding 5/6, 4/6, 1/6 will create an arithmetic sequence… 1, 5/6, 4/6, 1/2, 1/3, 1/6 – this conforms because the difference between successive numbers is constant (1/6) – thus allowing your 3 to belong to the same arithmetic sequence. NOTE – if you are not overly familiar with fractions, you may look at this sequence and determine that 1/3 and 1/2 are out of place – not so, 1/3 & 1/2 are the same fractions as 2/6 & 3/6 !
