Question

Find an explicit formula for the following sequence, an, which starts with a1=−4.

−4,1,6,11,16,…

Answer 1

`a_(n) = 5n-9`

Explanation:

we have the following sequence,

`-4,1,6,11,16,...`

where

`a_1=-4\\a_2=1\\a_3=6\\a_4=11\\a_5=16`

so

`a_2-a_1=1-(-4)=5` ----->
`a_2=a_1+5`

`a_3-a_2=6-1=5` ----->
`a_3=a_2+5`

`a_4-a_3=11-6=5` ----->
`a_4=a_3+5`

`a_5-a_4=16-116=5` ----->
`a_5=a_4+5`

Note there is a common difference d=5 between consecutive terms

This indicates the sequence is arithmetic with n th term

so

`a_(n) = a₁ + (n - 1)d`

where a₁ is the first term and d the common difference

substitute

`a_(n) = -4 + (n - 1)5`

`a_(n) = -4 +5n-5`

`a_(n) = 5n-9`