Question

Write an explicit formula for a1 = –2, an = an – 1 + 4, n ≥ 2.

Answer 1

The formula given is as follows

`a_(1)=-2, a_(n) = a_(n-1)+4, n\geq2`

Answer:

`a_(n) = 4n - 6`

Explanation:

Since it is given that
`n\geq   2`

`a_(2) = a_(1)+4`

`a_(3) = a_(2)+4`

... and so on...

we substitute
`a_(1)` to
`a_(2)` ,
`a_(2)` to
`a_(3)` and so on.. we will get

`a_(3) = a_(2) + 4`

`= a_(1) + 4 +4`

`= a_(1) + 2(4)`

`a_(4) = a_(3) + 4`

`= a_(1) + 2(4) +4`

`= a_(1) + 3(4)`

and the patterns goes on.

We can see from the recursion pattern that

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

`a_(n) = -2 + (n-1)(4)`

`= -2 + 4n - 4`

`= 4n - 6`