Arithmetic, Geometric, and Harmonic Sequences

Context: FIT1058_MOC · the three classic number-sequence families · each a one-line recurrence with a clean closed form · summed by Arithmetic Series / Geometric Series

Quick Revision

• 🎯 Objective: three standard sequence families ➔ arithmetic (diff $d$), geometric (ratio $r$), harmonic (reciprocals arithmetic).
• 📦 Core Components: closed forms $a+(n-1)d$, $a{r}^{n-1}$, $\frac{1}{a+(n-1)d}$.
• ⚡ Key Constraint: arithmetic grows linearly, geometric exponentially; harmonic defined via reciprocals.

📝 Core

1. The Three Families

• Arithmetic ➔ constant difference $d$ ➔ $f_n=a+(n-1)d$.
• Geometric ➔ constant ratio $r$ ➔ $f_n=a{r}^{n-1}$.
• Harmonic ➔ reciprocals are arithmetic ➔ $f_n=\frac{1}{a+(n-1)d}$.

2. Reading Off Parameters

• Arithmetic ➔ evens $0,2,4$ ($a=0,d=2$); countdown $10,\dots ,0$ ($a=10,d=-1$).
• Geometric ➔ powers of 2 ($a=1,r=2$).
• Harmonic ➔ canonical $(\frac{1}{n})$ (integers are arithmetic).

⚖️ Core Decision Matrix

Family	Growth	Sum
arithmetic	linear $\Theta (n)$	Arithmetic Series
geometric ($r>1$)	exponential	Geometric Series
geometric ($r<1$)	decays to 0	converges
harmonic	$\sim \ln n$ (partial sums)	harmonic number

When It Flips: the constant difference vs constant ratio is exactly why arithmetic is linear and geometric exponential — the same gap that makes geometric series converge while arithmetic ones diverge. Degenerate: $d=0$ or $r=1$ gives a constant sequence.

📊 Exam Execution Trace

Manual Execution Trace

Identify families:

Step / State	Sequence	Family	Params
0 (Init)	—	—	—
1	$3,7,11,15$	arithmetic	$a=3,d=4$
2	$2,6,18,54$	geometric	$a=2,r=3$
3	$3,4,6,12$	harmonic	recip. $d=-\frac{1}{12}$

⚠️ Common Mistakes

• 💡 Harmonic is indirect ➔ it is not itself arithmetic/geometric; prove harmonic facts by passing to reciprocals ($3,4,6,12$ → $\frac{4}{12},\frac{3}{12},\frac{2}{12},\frac{1}{12}$).

🧠 Active Recall

Give the recurrence and closed form for arithmetic and geometric sequences, and contrast their growth.

• Hint: Difference vs ratio.

Answer

• Short answer: Arithmetic $f_n=a+(n-1)d$ (linear); geometric $f_n=a{r}^{n-1}$ (exponential/decaying).
• Why: Constant $d$ vs $r$ ➔ additive step → linear, multiplicative step → exponential.

What makes a sequence harmonic, and why is $3,4,6,12$ one?

• Hint: Reciprocals arithmetic.

Answer

• Short answer: Reciprocals form an arithmetic sequence; $\frac{1}{3},\frac{1}{4},\frac{1}{6},\frac{1}{12}$ have common difference $-\frac{1}{12}$.
• Why: Canonical $(\frac{1}{n})$ ➔ integers are arithmetic.