That's exactly what the uniq command is made for:

NAME

       uniq - report or omit repeated lines



SYNOPSIS

       uniq [OPTION]... [INPUT [OUTPUT]]



DESCRIPTION

       Filter adjacent matching lines from INPUT (or standard input), writing to OUTPUT

       (or standard output).



       With no options, matching lines are merged to the first occurrence.

So

$ uniq test.txt 

a

-----

b

-----

c

-----

d

-----

e

-----

Alternatively, you can use this sed one-liner 69. Delete duplicate, consecutive lines from a file (emulates "uniq") from Sed One-Liners Explained, Part III: Selective Deletion of Certain Lines and Special Applications

sed '$!N; /^(.*)

1$/!P; D' test.txt

which might be preferred if you want to edit test.txt in place (by adding the -i or --in-place option).