On the first day of the play the auditorium was 1/3 full, on the second day it was it was 5/12 full, and on the third day it was 1/2 full. if the pattern continueshow full will it be on the fourth day

We have an aritmethic progression:

Nth=a₁ + (n-1) d

a₁=1/3

d=a₂-a₁ = (5/12) - (1/3) = (5-4) / 12=1/12

Therefore:

Nth=1/3 + (n-1) * 1/12

Nht=1/3+n/12 - (1/12)

Nth=n/12 + (4-1) / 12

Nth=n/12 + 3/12

Nth=n/12 + 1/4

Nth=n/12 + 1/4

To check:

Nth=1 ⇒1/12 + 1/4 = (1+3) / 12=4/12=1/3

Nth=2 ⇒2/12 + 1/4=1/6 + 1/4 = (2+3) / 12=5/12

Nth=3 ⇒3/12 + 1/4=1/4 + 1/4=2/4=1/2