For beginners, sequence problems can range anywhere from "simply fun" to "intimidatingly hard". While early stage mathletes (AMC 8 / 10) might find memorizing formula for arithmetic and geometric sequence useful, I'd recommend truly understanding the pattern in each sequence, as creative exploration of new patterns is the key to harder problems (AIME, for example)

• Arithmetic Sequence

  • Terms in Arithmetic Sequence:  Often, recognizing the pattern and finding the common difference is the challenge. Also, beware of the OBOE (Off By One Error) 

1985 AJHSME Problem 7

1988 AJHSME Problem 19

2022 AIME I Problem 6

• • Sum of Arithmetic Sequence (Series)

1985 AJHSME Problem 2

1987 AJHSME Problem 3

1988 AJHSME Problem 4

1990 AJHSME Problem 11

1993 AJHSME Problem 19

1994 AJHSME Problem 2 Problem 17

• • Simple "shortcuts": these facts can help you solve an arithmetic sequence problem much faster:

    • The average of an arithmetic sequence is simply its median

    • The average of an arithmetic sequence is also average of the highest and lowest term

1986 AJHSME Problem 25

2021 AIME II Problem 1

2023 AIME II Problem 1

• Geometric Sequence

1991 AJHSME Problem 25

2020 AIME I Problem 8

2020 AIME I Problem 2

• Sequence of Other Patterns

  • Tip: the general strategy here is to try a few terms to get the pattern, prove that the pattern holds (optional depending on the nature of the competition and your time), then develop a solution based on the pattern. Paul Zeitz has a great discussion about this mindset in Example 1.2.1 in his book "The Art and Craft of Problem Solving". 

1990 AJHSME Problem 21

1991 AJHSME Problem 17

1992 AJHSME Problem 15

1993 AJHSME Problem 24

1994 AJHSME Problem 15

2020 AIME II Problem 6 Problem 8

2021 AIME I Problem 10

• Telescoping

1985 AJHSME Problem 9