Question

How do you write a rule for the nth term of the arithmetic sequence `a_6=-31, a_14=-135`?

Answer 1

`T_n = -13n +47`

Explanation:

For any term in an AP, the rule is the same: `T_n = a+(n-1)d`

For the `6th` term we have..
`T_6 = a+ 5d =-31" "larr (6-1=5)`

For the `14th` term we have:
`T_14 = a+13d =-135" "larr(14-1 =13)`

To be able to write the rule we need to find `a` and `d`.
Solve the equations simultaneously:

`color(white)(xxxxxx)T_6 = a+" "5d=-31`.........................................A
`color(white)(xxxxxx)T_14 = a+ " "13d=-135`.........................................B

A-B:`color(white)(wwwwwwwwww)-8d = 104`
A-B:`color(white)(wwwwww.wwwwww)d = -13`

Subst in A:`rarra +5(-13) = -31`
`" "a" " -65 = -31`
`" "a " "= -31+65`
`" "a " "= 34`

Now we know that the first term is `34` and that `d=-13`

The first `6` terms are: `34, " "21," " 8, -5, -18, -31 .....`

The rule for the `nth` term is:

`T_n = 34+(n-1)(-13)`

`T_n = 34-13n +13`

`T_n = -13n +47`