🔢 Series & AP Shortcuts

If every jump is the same size, the small values and the large values balance perfectly around the centre. So:

$\text{average}=\dfrac{\text{first}+\text{last}}{2}=\text{middle term}$

This one rule covers consecutive integers, even numbers, odd numbers and multiples — the count of terms is irrelevant. For odd-length lists the average is literally the centre value; for even-length lists it sits halfway between the two middle ones.

Memorise these instant results: first $n$ natural numbers $\to\dfrac{n+1}{2}$  ·  first $n$ odd $\to n$  ·  first $n$ even $\to n+1$  ·  first $n$ multiples of $k$ $\to\dfrac{k(n+1)}{2}$.

The exception — squares. Squares like $1,4,9,16,\dots$ are not evenly spaced, so the middle trick fails. Use $\dfrac{(n+1)(2n+1)}{6}$ for the average of the first $n$ squares.