Question

The recursive rule for a sequence is shown.an=an−1+9a1=21 what is the explicit rule for this sequence?

Answer 1

Answer:

Meaning of Explicit rule , is the general rule for a sequence.

The given sequence is defined as,

$a_{n}=a_{n-1}+9a_{1}-21$----(1)

Common Difference(d) of an airthmetic Sequence= N th term - (N-1) th term

                                 $=a_{n}-a_{n-1}\\\\=a_{n-1}-a_{n-2}$                                            

Equation 1 can be written as,

$a_{n}-a_{n-1}=9 a_{1}-21\\\\ d =9a_{1}-21$

General rule or explicit rule for this sequence is given as:

$a_{n}=a_{1} +(n-1)d\\\\a_{n}=a_{1}+(n-1)(9a_{1}-21)\\\\a_{n}=a_{1}+9na_{1}-21 n-9a_{1}+21\\\\a_{n}=-8a_{1}+21+n(9a_{1}-21)$