**Given:**

The sequence is
an=(−1)nnn!+1
. (1)

**Calculation:**

Obtain the first term of the sequence.

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

a1=(−1)1⋅11!+1 =(−1)⋅11+1 =−12

Thus, the first term of the sequence is
a1=−12_
.

Obtain the second term of the sequence.

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

a2=(−1)2⋅22!+1 =1⋅22+1 =23

Thus, the second term of the sequence is
a2=23_
.

Obtain the third term of the sequence.

Substitute 3 for *n* in equation (1).

a3=(−1)3⋅33!+1 =(−1)⋅36+1 =−37

Thus, the third term of the sequence is
a3=−37_
.

Obtain the fourth term of the sequence.

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

a4=(−1)4⋅44!+1 =1⋅424+1 =425

Thus, the fourth term of the sequence is
a4=425_
.

Obtain the fifth term of the sequence.

Substitute 5 for *n* in equation (1).

a5=(−1)5⋅55!+1 =(−1)⋅5120+1 =−5121

Thus, the fifth term of the sequence is
a5=−5121_
.

Therefore, the first five terms of the sequence are
−12,23,−37,425,−5121
.