Question

What is the explicit rule for the sequence `-1, -6, -11, -16`?

Answer 1

The rule is -5n+4

Explanation:

The difference between any of these terms is -5. So it is an Arithmetic sequence.

Let n be the sequence location count

Then we have

`" "n" sequence value"`
`" "1" -1"`
`" "2" -6"`
`" "3" -11"`
`" "4" -16"`

To develop a rule we have to relate the sequence value to its relevant number 'n'. In that we need something like:

`"First term + (difference"xxn)`

We know that the difference is always -5 so lets see what happens

`" "n" sequence value -1+(-5)n"`
`" "1" -1 -6 "color(red)("Failed")`
`" "2" -6"`
`" "3" -11"`
`" "4" -16"`

We can adjust the is result back to where it should be by adding 5. So lets see if that works!

But -1+(-5)n +5 = (-5)n+4

`" "n" sequence value (-5)n +4"`
`" "1" -1 -1 "`
`" "2" -6 -6 "`
`" "3" -11 -11 "`
`" "4" -16 -16 "`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
That worked! So the rule is -5n+4