Question

1/6,1/3,1/2,2/3,___, ___ (hint: rewrite each fraction so they all have a common deniminator of 6) (write the next 2 terms and write the rule)

Answer 1

`\textsf{The next two terms are:}\quad (5)/(6)\;\textsf{and}\;1`

`\textsf{The rule for the $n$th term is:}\quad a_n=(n)/(6)`

Explanation:

Given sequence:

`(1)/(6),\;\;(1)/(3),\;\;(1)/(2),\;\;(2)/(3),\;...`

To find the next two terms in the given sequence, begin by rewriting each fraction so that they all have a common denominator of 6. To achieve this, multiply the numerator and denominator of each fraction by the same number needed to make the denominator equal to 6:

`(1)/(6),\;\;(1\cdot 2)/(3\cdot 2),\;\;(1\cdot 3)/(2\cdot 3),\;\;(2\cdot 2)/(3\cdot 2),\;...`

`(1)/(6),\;\;(2)/(6),\;\;(3)/(6),\;\;(4)/(6),\;...`

Notice that the numerator of each term is the position of the term.

Therefore, the next two terms (a₅ and a₆) of the sequence are:

`a_5=(5)/(6)`

`a_6=(6)/(6)=1`

Therefore, the rule for the nth term is:

`a_n=(n)/(6)`