Question

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

Answer 1

Answer:

$a_{n}=45-3n$

Step-by-step 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

$a_{n}=45-3n$