Step 1: Identify the type

• Constant difference -> AP
• Constant ratio -> GP
• Reciprocals in AP -> HP
• Product of AP and GP -> AGP
• Fraction with products in denominator -> Telescoping

Step 2: For finding terms, assume symmetric form

• 3 in AP: a-d, a, a+d (sum=3a, use this to find a)
• 3 in GP: a/r, a, ar (product=$a^3$, use this to find a)
• 4 in AP: a-3d, a-d, a+d, a+3d (common diff = 2d)

Step 3: For sums, decompose the general term

• Polynomial -> standard sum formulas
• Fraction -> partial fractions + telescope
• Mixed AP*GP -> S-rS technique

Step 4: For optimization, use AM-GM

• Verify all quantities are positive
• Match the decomposition to the target expression
• Check equality condition

Step 5: Cross-check with small values

• Verify your formula at n=1, n=2
• Ensure $S_1$ = $a_1$

Common errors to avoid:

• nth term uses (n-1), not n
• GP sum formula needs r != 1
• Infinite GP needs |r| < 1
• $S_n$ quadratic with constant term != AP
• AM-GM needs positive quantities