Explain when I'd use each equation and provide example questions for each formula please

Sequence and Series Formulas: Given: an represents the value of nth term Sn represents the sum of first n terms So represents the sum of an infinite geometric series represents the common ratio d represents the common difference Arithmetic Geometric a = a + (n-1)d a = arn-1 a, = an-1 + d a = an-1 . r S, = = ( 1 + a, ) S = (1-r ) (1-r) . = 1 a1 2 S = 1 2a + (n -1)d Soo = ( 1 - r ) ' r