Answer:
The recursive formula is given as:
[tex]a_n = 3a_{n-1}+2[/tex] , n is the number of terms
where,
[tex]a_4 = 20[/tex]
We have to find first three terms of the sequence.
For n =4 ;
[tex]a_4 = 3a_{4-1}+2[/tex]
⇒[tex]a_4 = 3a_3+2[/tex]
⇒[tex]20 = 3a_3+2[/tex]
Subtract 2 from both sides we have;
[tex]18= 3a_3[/tex]
Divide both sides by 3 we have;
[tex]6 = a_3[/tex]
or
[tex]a_3 =6[/tex]
For n = 3;
[tex]a_ 3= 3a_{2}+2[/tex]
⇒[tex]6 = 3a_2+2[/tex]
Subtract 2 from both sides we have;
[tex]4= 3a_2[/tex]
Divide both sides by 3 we have;
[tex]a_2 = \frac{4}{3}[/tex]
For n =2
[tex]a_2= 3a_{1}+2[/tex]
⇒[tex]\frac{4}{3} =3a_1+2[/tex]
Subtract 2 from both sides we have;
[tex]-\frac{2}{3}= 3a_3[/tex]
Divide both sides by 3 we have;
[tex]a_1 =-\frac{2}{9}[/tex]
Therefore, the first three terms of the sequence are:
[tex]a_1 =-\frac{2}{9}[/tex]
[tex]a_2 = \frac{4}{3}[/tex]
[tex]a_3 =6[/tex]
