grep (bash) multi-line pattern

Question

In bash (4.3.46(1)) I have some multi-line so called fasta records where each record is initiated by on line with >name and the following lines DNA sequence ([AGCTNacgtn]),

Answer 1

You can use awk with custom RS:

awk -v n=2 -v RS='(^|\n)>' 'NR==n+1{print ">" $0}' file    
>chr2
TTGNACACCC
TGGGGGAGTA