Lesson 17 — Arithmetic progressions (PA)

Definition and formulas

General term

$a_n = a_1 + (n-1) r$

Proof by induction on $n$. Equivalent to $a_n = a_p + (n-p) r$ for any $p$.

Sum of the first $n$ terms

$S_n = \sum_{k=1}^n a_k = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)r)}{2}$

Proof (Gauss as a child, ~1789): write $S_n$ twice, once in increasing order and once in decreasing order:

$\begin{aligned} S_n &= a_1 + a_2 + \ldots + a_n \\ S_n &= a_n + a_{n-1} + \ldots + a_1 \end{aligned}$

Adding term by term: $2 S_n = n(a_1 + a_n)$, since each pair sums to $a_1 + a_n$. ∎

Properties

• Arithmetic mean: three consecutive terms satisfy $a_n = (a_{n-1} + a_{n+1})/2$.
• Increasing if $r > 0$, decreasing if $r < 0$, constant if $r = 0$.

Numerical example

PA with $a_1 = 5$, $r = 3$: terms $5, 8, 11, 14, 17, \ldots$

$a_{20} = 5 + 19 \cdot 3 = 62$. $S_{20} = 20 \cdot (5 + 62)/2 = 670$.