In​ March, a family starts saving for a vacation they are planning for the end of August. The family expects the vacation to cost ​$1483. They start with ​$120. At the beginning of each month they plan to deposit 25​% more than the previous month. Will they have enough money for their​ trip? If​ not, how much more do they​ need?

No, They will need 131.94 more dollars.
The exponential growth formula can help us find how much is to be deposited each month and then adding them together for the 5 months after the initial deposit will give us the amount saved.
[tex]y=a(1+r)^{x} [/tex]
a is the initial value (120)
r is the rate of growth (0.25)
x is time [0,1,2,3,4,5)
March
[tex]120(1+.25)^{0} = 120[/tex]
April
[tex]120(1+.25)^{1} = 150[/tex]
May
[tex]120(1+.25)^{2} = 187.5[/tex]
June
[tex]120(1+.25)^{3} = 234.38[/tex]
July 
[tex]120(1+.25)^{4} = 292.97 [/tex]
August
[tex]120(1+.25)^{5} = 366.21[/tex]

Totals
120+150+187.5+234.38+292.97+366.21= 1351.06 totalt saved by august

1483-1351.06 = 131.94 needed.