The n th term of the sequence is given as,
a n =3 a n−1 +2(1)
And it is given that a 1 =3.
Substitute n=2 in equation (1),
a 2 =3 a 2−1 +2 =3 a 1 +2 =3( 3 )+2 =11
Substitute n=3 in equation (1),
a 3 =3 a 3−1 +2 =3 a 2 +2 =3( 11 )+2 =35
Substitute n=4 in equation (1),
a 4 =3 a 4−1 +2 =3 a 3 +2 =3( 35 )+2 =107
Substitute n=5 in equation (1),
a 5 =3 a 5−1 +2 =3 a 4 +2 =3( 107 )+2 =323
The required first five terms are 3,11,35,107,323.
The corresponding series will be,
3+11+35+107+323+⋯