Change in arithmetic and geometric sequences

AP Precalculus· AP Precalculus CED — Exponential and Logarithmic Functions· 14 min read

1. Core Concepts of Discrete Sequence Change★★☆☆☆⏱ 3 min

Sequences are discrete functions with domains restricted to subsets of integers, so analyzing change follows the same logic as continuous functions but adapted to discrete inputs. This topic makes up 3–5% of the total AP Precalculus exam score, appearing in both multiple-choice and free-response sections. It acts as a critical bridge between linear change (Unit 1) and the exponential change that forms the core of Unit 2.

📘 Definition

Arithmetic Sequence

A discrete sequence defined by constant additive change between consecutive terms. The constant change is called the common difference $d$ .

Example:

$1, 4, 7, 10, ...$ with common difference $d = 3$

📘 Definition

Geometric Sequence

A discrete sequence defined by constant proportional (multiplicative) change between consecutive terms. The constant ratio is called the common ratio $r$ .

Example:

$2, 6, 18, 54, ...$ with common ratio $r = 3$

2. Change in Arithmetic Sequences★★☆☆☆⏱ 4 min

The sequence difference operator gives the additive change between consecutive terms, defined as:

\Delta a_n = a_{n+1} - a_n

By definition, for any arithmetic sequence, $Δ a_{n} = d$ (constant) for all $n$ . This constant consecutive change is the defining property of arithmetic sequences. The explicit form of an arithmetic sequence starting at $n = 0$ is $a_{n} = a_{0} + n d$ , and average rate of change over any interval from $n = k$ to $n = m$ is always constant:

\frac{a_m - a_k}{m - k} = d

📐 Worked Example

Given the sequence defined explicitly by $a_{n} = 7 - 4 n$ , find (1) the total change between the 3rd and 8th term, and (2) the average rate of change over the interval $n = 1$ to $n = 5$ , then confirm it matches the common difference.

1
First confirm the sequence is arithmetic: it is linear in $n$ , so the common difference is the coefficient of $n$ , $d = - 4$ .
2
Calculate the terms needed: $a_{3} = 7 - 4 (3) = - 5$ , $a_{8} = 7 - 4 (8) = - 25$ .
3
Total change = $a_{8} - a_{3} = - 25 - (- 5) = - 20$ . Verify with the change formula: total change = $d (8 - 3) = - 4 (5) = - 20$ , which matches.
4
Calculate average rate of change from $n = 1$ to $n = 5$ : $a_{1} = 7 - 4 (1) = 3$ , $a_{5} = 7 - 4 (5) = - 13$ . Average rate of change is:
5
$\frac{a_5 - a_1}{5-1} = \frac{-13 - 3}{4} = -4$
6
The average rate of change matches the common difference $d = - 4$ , as expected for an arithmetic sequence.

Exam tip:

When a problem gives you non-consecutive terms of an arithmetic sequence, calculate the common difference directly as $d = \frac{a _{j} - a _{i}}{j - i}$ without solving for the initial term first, to save time on MCQs.

3. Change in Geometric Sequences★★★☆☆⏱ 4 min

Geometric sequences are discrete exponential sequences defined by a constant common ratio $r$ , the constant proportional change between consecutive terms. By definition, the ratio of any two consecutive terms is constant:

\frac{a_{n+1}}{a_n} = r \quad \text{for all } n

Unlike arithmetic sequences, the additive change between consecutive terms of a geometric sequence is not constant:

\Delta a_n = a_{n+1} - a_n = a_n(r-1)

Additive change is proportional to the current term value, which is the discrete equivalent of the continuous exponential property that the derivative of an exponential function is proportional to the function's value. For non-consecutive terms separated by $k$ positions, the proportional change is constant:

\frac{a_{n+k}}{a_n} = r^k

The explicit form for a sequence starting at $n = 0$ is $a_{n} = a_{0} r^{n}$ , or $a_{n} = a_{1} r^{n - 1}$ for sequences starting at $n = 1$ . Unlike arithmetic sequences, the average rate of change over an interval of a geometric sequence is not constant.

📐 Worked Example

A geometric sequence has $a_{2} = 18$ and $a_{5} = 486$ , with all positive terms. Find the common ratio $r$ , then calculate the additive change between $a_{3}$ and $a_{6}$ .

1
Use the proportional change property for non-consecutive terms: $\frac{a _{5}}{a _{2}} = r^{5 - 2} = r^{3}$ .
2
Substitute the given values: $\frac{486}{18} = 27 = r^{3}$ , so $r = 327 = 3$ (we discard the negative root because all terms are positive).
3
Find the required terms: $a_{3} = a_{2} \cdot r = 18 \cdot 3 = 54$ , $a_{6} = a_{5} \cdot r = 486 \cdot 3 = 1458$ .
4
Calculate additive change: $Δ a = a_{6} - a_{3} = 1458 - 54 = 1404$ .
5
Verify with the proportional change rule: $Δ a = a_{3} (r^{3} - 1) = 54 (27 - 1) = 54 \cdot 26 = 1404$ , which confirms the result.

Exam tip:

If the problem does not specify all terms are positive, remember that $r^{k} = c$ (for positive $c$ and even $k$ ) has two solutions: $r = k c$ and $r = - k c$ . Do not forget the negative solution unless explicitly told to rule it out.

4. Comparing Long-Term Growth★★★★☆⏱ 3 min

A key AP Precalculus skill is comparing the long-term behavior of increasing arithmetic (discrete linear) and increasing geometric (discrete exponential) sequences as $n$ grows large. For any increasing arithmetic sequence ( $d > 0$ ) and any increasing geometric sequence ( $r > 1$ ), exponential growth will always outpace linear growth for sufficiently large $n$ , even if the arithmetic sequence is larger for small values of $n$ . This question often requires testing integer values (or using logarithms) to find the minimum $n$ where the geometric sequence surpasses the arithmetic sequence.

📐 Worked Example

Arithmetic sequence $A (n) = 100 + 20 n$ and geometric sequence $G (n) = 50 (1.12)^{n}$ are defined for all integers $n \geq 0$ . What is the minimum integer $n$ such that $G (n) > A (n)$ ?

1
Set up the inequality we need to solve: $50 (1.12)^{n} > 100 + 20 n$ .
2
Test small values to identify the general trend: at $n = 0$ , $G = 50 < 100 = A$ ; at $n = 15$ , $G \approx 273.68 < 400 = A$ ; at $n = 25$ , $G \approx 850 > 600 = A$ .
3
Narrow down by checking smaller values: at $n = 20$ , $G (20) \approx 482.3$ , $A (20) = 100 + 20 (20) = 500$ , so $G (20) < A (20)$ .
4
Check $n = 21$ : $G (21) = 482.3 (1.12) \approx 540.2$ , $A (21) = 100 + 20 (21) = 520$ , so $G (21) > A (21)$ .
5
The minimum integer $n$ is 21, since $n = 20$ still has $G (n) < A (n)$ .

Exam tip:

When asked for the minimum $n$ where a geometric sequence exceeds an arithmetic sequence, always check the integer one below your candidate value. Exams regularly include the candidate value one above the correct answer as a MCQ distractor.

5. Common Pitfalls

Wrong move:

Treating the common ratio of a geometric sequence as the additive change between consecutive terms, e.g. writing $Δ a_{n} = r$ instead of $a_{n} (r - 1)$

Why:

Students confuse the defining multiplicative property of geometric sequences with the question's request for 'change', which is always additive unless explicitly stated otherwise

Correct move:

Underline whether the question asks for 'change' (additive) or 'common ratio' (multiplicative) before starting to solve

Wrong move:

Calculating the common difference of an arithmetic sequence as $a_{j} - a_{i}$ for non-consecutive terms, e.g. for $a_{2} = 5$ , $a_{5} = 14$ , calculating $d = 9$

Why:

Students are used to finding $d$ from consecutive terms, so they forget to divide by the gap in indices

Correct move:

Always divide the difference in term values by the difference in term positions to get $d$ , regardless of whether terms are consecutive

Wrong move:

Stopping at the first crossing of $G (n) > A (n)$ and assuming $G (n)$ stays larger, even when $0 < r < 1$

Why:

Students forget that geometric sequences with $0 < r < 1$ decay to zero, so they can exceed an increasing arithmetic sequence for small $n$ then fall behind permanently

Correct move:

Check the value of $r$ first: if $r < 1$ and $A (n)$ is increasing, note that $G (n)$ will eventually fall back below $A (n)$

Wrong move:

Off-by-one errors when using $a_{n} = a_{1} + (n - 1) d$ or $a_{n} = a_{1} r^{n - 1}$ , e.g. calculating the 5th term as $a_{1} r^{5}$

Why:

Confusion between sequences starting at $n = 0$ vs $n = 1$ is extremely common

Correct move:

Write down explicitly whether the sequence starts at $n = 0$ or $n = 1$ before plugging into any explicit formula

Wrong move:

Calculating total change for an arithmetic sequence over $k$ steps as $d$ instead of $k d$

Why:

Students confuse consecutive change (change over 1 step) with change over multiple steps

Correct move:

Always multiply the common difference by the difference in indices to get total change for arithmetic sequences

6. Quick Reference Cheatsheet

Category	Formula	Notes
Arithmetic consecutive change	$Δ a_{n} = a_{n + 1} - a_{n} = d$	Constant for all $n$ , defining property of arithmetic sequences
Total change (arithmetic, non-consecutive)	$a_{j} - a_{i} = d (j - i)$	Works for any $i < j$
Average rate of change (arithmetic)	$\frac{a _{j} - a _{i}}{j - i} = d$	Always constant, equal to common difference
Geometric common ratio	$\frac{a _{n + 1}}{a _{n}} = r$	Constant for all $n$ , defining property of geometric sequences
Consecutive additive change (geometric)	$Δ a_{n} = a_{n} (r - 1)$	Proportional to current term, not constant
Ratio of non-consecutive terms (geometric)	$\frac{a _{j}}{a _{i}} = r^{j - i}$	Works for any $i < j$
Explicit form ( $n = 0$ start)	Arithmetic: $a_{n} = a_{0} + d n$ ; Geometric: $a_{n} = a_{0} r^{n}$	Adjust exponent/offset to $(n - 1)$ for $n = 1$ start
Long-term growth comparison	For $d > 0, r > 1, a_{0} > 0$ : $r^{n}$ outgrows $d n$ as $n \to \infty$	Exponential growth always outpaces linear growth for large $n$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · MCQ
Find average rate of change of sequence
2024 · FRQ
Compare growth of two sequences

What's Next

This topic is the foundational introduction to discrete exponential change, the core of AP Precalculus Unit 2: Exponential and Logarithmic Functions. Mastering the difference between additive constant change (arithmetic/linear) and multiplicative constant change (geometric/exponential) is critical for all subsequent topics in this unit. You will next extend these discrete properties to continuous exponential functions, where the multiplicative change property translates to the key rule that the rate of change of an exponential function is proportional to the function's value. This topic also lays the groundwork for geometric series, used for applications like compound interest and annuity calculations, and for comparing linear and exponential models in real-world contexts.