Question

How do you find the nth term rule for `1/2,1,3/2,2,...`?

Answer 1

`nth. =1/2n`

Explanation:

`1/2,1,3/2,2,...`

firstly see what the terms increase by

difference `d=1-1/2=1/2`

so nth term is of the form

#nth.=1/2n+c#

use the first term and `n=1` to evaluate `c`

`1/2=1/2xx1+c`

`=>c=0`

`:.nth. =1/2n`

a quick mental check on the next few terms confirms the result.

Answer 2

`T_n = 1/2n`

Explanation:

The sequence might be easier to recognise as an AP if the terms are written as improper fractions:

`T_n = 1/2, " "1," "1 1/2," " 2," " 2 1/2," " 3" "...`

An even better way to write this is in halves:

`T_n = 1/2, " "2/2," " 3/2, " "4/2," " 5/2...`

`" "n = 1, " " 2 ," " 3 , " "4 , " "5 .....`

It is then immediately obvious that each term is just `n/2`

Using the formula/ general rule:

The nth term is written as `T_n = a + d(n-1)`

All the values you need are available from the sequence.

`a = 1/2, " " d = 1/2" "` substitute into `T_n = a + d(n-1)`

`T_n= 1/2 + 1/2(n-1)" "larr` multiply out and simplify

`T_n = 1/2+1/2n-1/2`

`T_n = 1/2n`