Topic 2.1 Notes – Change in Arithmetic and Geometric Sequences

Two patterns a sequence can follow

A sequence is a function whose inputs are whole numbers and whose outputs are real numbers. If you see $a_n$ or $g_n$, that subscript $n$ is the index or term number. It is not multiplication.

• $a_0$ means the term at index 0
• $a_5$ means the term at index 5
• Same idea for $g_n$

A sequence is graphed as discrete points $(n,\text{term})$, because only whole-number inputs belong in the domain.

Arithmetic sequence

An arithmetic sequence changes by repeated addition. The amount added each step is the common difference $d$.

• “increases by 7 each step” ⟶ arithmetic
• “loses 12 per cycle” ⟶ arithmetic

For arithmetic sequences:

• $d>0$ increasing
• $d<0$ decreasing
• $d=0$ constant

Geometric sequence

A geometric sequence changes by repeated multiplication. The multiplier is the common ratio $r$.

• “multiplied by 1.4” ⟶ geometric
• “doubles” ⟶ geometric
• “retains 85\%” ⟶ geometric with $r=0.85$

For positive geometric sequences:

• $r>1$ increasing
• $0<r<1$ decreasing
• $r=1$ constant

One idea students mix up a lot: an increasing arithmetic sequence adds the same amount each time, but an increasing positive geometric sequence grows by bigger and bigger amounts even though the ratio stays the same.

Writing the General Term

The rule for a sequence depends on where indexing begins.

Arithmetic forms

$$a_n=a_0+dn$$

$$a_n=a_k+d(n-k)$$

Here, $n-k$ means the number of index steps from term $k$ to term $n$.

Geometric forms

$$g_n=g_0r^n$$

$$g_n=g_kr^{\,n-k}$$

Again, $n-k$ counts steps. If it’s negative, you’re moving backward.

If the first listed term is index 1, use:

• $a_n=a_1+d(n-1)$
• $g_n=g_1r^{n-1}$

That $(n-1)$ is a classic quiz mistake. Count steps between terms, not just the labels.

Finding the Change Parameter and Building the Rule

From a table with consecutive indices:

• arithmetic ⟶ subtract consecutive terms to get $d$
• geometric ⟶ divide consecutive terms to get $r$ if terms are nonzero

From two nonconsecutive terms:

• arithmetic

  $$d=\frac{a_j-a_k}{j-k}$$

• geometric

  $$\frac{g_j}{g_k}=r^{\,j-k}$$

If indices skip by more than 1, the displayed change is over multiple steps.

• arithmetic ⟶ divide by number of steps
• geometric ⟶ take the appropriate root

Examples you should know:

• Auditorium rows: 31 seats at row 4 and 46 seats at row 9

  $$d=\frac{46-31}{9-4}=3,\qquad a_n=31+3(n-4)$$

• Lab culture mass: 72 mg at hour 1 and 162 mg at hour 3

  $$\frac{162}{72}=2.25=r^2 \Rightarrow r=1.5,\qquad g_n=72(1.5)^{n-1}$$

How to Tell Which Type You Have

• Arithmetic if consecutive differences are constant
• Geometric if consecutive ratios are constant and the needed terms are nonzero
• Neither if neither test works

If zeros appear, division may fail, so check whether each term is found by multiplying the previous one by the same ratio.

A nonzero constant sequence is both:

• arithmetic with $d=0$
• geometric with $r=1$

Edge cases:

• negative $r$ gives alternating signs, so it usually is not increasing or decreasing throughout
• $r=0$ makes later terms zero, so the usual ratio test breaks

Common Mistakes on the Exam

• Connecting sequence points like a continuous graph
• Mixing up add the same amount with multiply by the same factor
• Using a multi-step change as if it were the one-step $d$ or $r$
• Forgetting the root in geometric problems when indices differ by more than 1
• Confusing $a_0$ with $a_1$, or missing the $(n-k)$ step count
• Assuming geometric growth is automatically larger than arithmetic growth at first
• Using series formulas here. This topic is about terms, not sums.