The area of the wall is #22*27=594# (square inches).
Define the area Lucy completed on #n#th day to be #a_n# (square inches).
Since #a_1=48# and the sequence is arithmetic, #a_n# can be expressed as #a_n=48+(n-1)d#, where #d# is the common difference.
She completed the wall hanging in 27 days.
The sum of #a_n# for #n=1# to #27# is
#S=sum_(k=1)^27a_k=1/2*27*(a_1+a_27)#
#=1/2*27*{48+(48+26d)}#
#=1296+351d#
Therefore, the following equation will give the difference #d#.
#1296+351d=594#
#351d=-702#
#d=-2#
The area Lucy completed on #n#th day is #a_n=48+(n-1)*(-2)=50-2n#.
However, there is a problem.
#a_n=50-2n# gives negative value for #n=26,27#.
#a_26=-2, a_27=-4#
Actually, she had completed the work on #22#th day since #a_23+a_24+a_25+a_26+a_27=4+2+0+(-2)+(-4)=0#.
