Question

10. An arithmetic sequence has this recursive formula. What is the explicit formula?​

Answer 1

`a_(n)=45-3n`

Explanation:

Method 1:

Arithmetic sequence is in the form

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

d is the common difference, can be found by:

`d=a_(n)-a_(n-1)=-3`

Subtituting the
`a_(1)` and
`d`

You get:

`a_(n)=42+(-3)(n-1)=45-3n`

Method 2 (Mathematical induction):

Assume it is in form
`a_(n)=45-3n`

Base step:
`a_(1) =45-3(1)=42`

Inducive hypophesis:
`a_(n)=45-3n`

GIven:
`a_(n+1) =a_(n)-3`

`a_(n+1)=45-3n-3=45-3(n+1)`

Proved by mathematical induction

`a_(n)=45-3n`