**Given:**

The sequence is
an+1=5an−3
. (1)

The first term of the sequence is
a1=1
.

**Calculation:**

Obtain the second term of the sequence.

Substitute 1 for *n* in equation (1).

a1+1=5a1−3

a2=5a1−3
(2)

Substitute 1 for
a1
in equation (2).

a2=5⋅1−3=2

Thus, the second term of the sequence is
a2=2_
.

Obtain the third term of the sequence.

Substitute 2 for *n* in equation (1).

a2+1=5a2−3

a3=5a2−3
(3)

Substitute 2 for
a2
in equation (3).

a3=5⋅2−3=10−3=7

Thus, the third term of the sequence is
a3=7_
.

Obtain the fourth term of the sequence.

Substitute 7 for *n* in equation (1).

a3+1=5a3−3

a4=5a3−3
(4)

Substitute 3 for
a3
in equation (4).

a4=5⋅7−3=35−3=32

Thus, the fourth term of the sequence is
a4=32_
.

Obtain the fifth term of the sequence.

Substitute 4 for *n* in equation (1).

a4+1=5a4−3

a5=5a4−3
(5)

Substitute 32 for
a4
in equation (4).

a5=5⋅32−3=160−3=157

Thus, the fifth term of the sequence is
a5=157_
.

Therefore, the first five terms of the sequence are
1,2,7,32,157_
.