Question

Consider a sequence given by the formula an = an−1 − 5, where a1 = 12 and n ≥ 2.

b. Write an explicit formula.

Answer 1

`a_(n)  = 12 + (n - 1) (-5)`

Explanation:

An explicit formula has two parts in it.

1) The initial value i.e.
`a_(1)  = 12`

2) The common difference between consecutive terms

To find the common difference we will use the first 5 terms which are as follow

12, 7, 2, -3, -8

the common difference is of -5

The general formula for an explicit arithmetic equation is as follow

`a_(n) = a_(1) + (n-1) d\\`

(This is an arithmetic sequence since it has a common difference)

Where

`a_(1)` is the initial value

`n` is the nth term

`d` is the common difference

hence the explicit formula becomes

`a_(n)  = 12 + (n - 1) (-5)`