Arithmetic vs geometric sequence: the sorting table

The sorting table

	Arithmetic	Geometric
Relation	$u_{n+1} = u_n + r$	$u_{n+1} = q \cdot u_n$
Name	Common difference $r$	Common ratio $q$
Recognize	$u_{n+1} - u_n =$ constant	$u_{n+1} / u_n =$ constant
Formula	$u_n = u_0 + n r$	$u_n = u_0 \cdot q^n$
Sum of $n+1$ terms	$(n+1) \cdot \dfrac{u_0 + u_n}{2}$	$u_0 \cdot \dfrac{1 - q^{n+1}}{1 - q}$ (if $q \neq 1$)

Recognize in 3 seconds

Compute $u_1 - u_0$ and $u_2 - u_1$:

• Equal ⇒ arithmetic, common difference = this value.

Compute $u_1 / u_0$ and $u_2 / u_1$:

• Equal ⇒ geometric, common ratio = this value.

Neither ⇒ neither arithmetic nor geometric.

Example 1: arithmetic

$3, 7, 11, 15, 19, \ldots$

$7 - 3 = 4$, $11 - 7 = 4$ ⇒ arithmetic, common difference $4$.

$u_n = 3 + 4n$.

Example 2: geometric

$2, 6, 18, 54, \ldots$

$6/2 = 3$, $18/6 = 3$ ⇒ geometric, common ratio $3$.

$u_n = 2 \cdot 3^n$.

Monotonicity

Arithmetic: sign of $r$ decides.

• $r > 0$ ⇒ increasing
• $r < 0$ ⇒ decreasing

Geometric ($u_0 > 0$): sign of $q - 1$ decides.

• $q > 1$ ⇒ increasing
• $0 < q < 1$ ⇒ decreasing
• $q < 0$ ⇒ sequence alternates in sign

Common mistakes

1. Confusing arithmetic and geometric common ratio.
2. Dividing by $u_0 = 0$ to check geometric.
3. Wrong sum formula.
4. Writing $u_n = u_1 + n r$ instead of $u_1 + (n-1) r$ (starting index).

?À toi de jouer

The sequence 1, 3, 9, 27, 81, ... is:

A.Geometric, ratio 3B.Arithmetic, difference 2C.Geometric, ratio 2D.Neither arithmetic nor geometric

Valider