Overview

Sequences appear across AMC and AIME. Recognize standard forms and convert between recursive and closed forms.

Key Ideas

• Arithmetic sequence: $a_n=a_1+(n-1)d$.
• Geometric sequence: $a_n=a_1 r^{n-1}$.
• Telescoping sums simplify when terms cancel in pairs.

Worked Example

If $a_1=3$ and $a_{n+1}=2a_n+1$, compute $a_3$.

Then $a_2=2\cdot3+1=7$, and $a_3=2\cdot7+1=15$.

Practice Problems