Standard PMI results: sums, divisibility, inequalities (First n naturals sum to n times n plus one over two; divisibility claims)
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First n naturals sum to n times n plus one over two; divisibility claims often expand f(k+1) minus f(k) and use the induction hypothesis mod a; inequalities use the bound from P(k) and one extra inequality to reach P(k+1).
What each symbol means
| Symbol | What it stands for |
|---|---|
| n | Positive integer index |
| f(n) | Integer expression in divisibility problems |
| g(n) | Real expression in inequality induction |
When to use this
Sum formulas need matching lower index (usually 1); divisibility needs integer f(n); inequalities need a usable comparison (e.g. AM-GM, Bernoulli, or monotonicity).