Change in Arithmetic and Geometric Sequences
Why this matters
Imagine you land two different summer job offers in Chicago.
The first is at a cool coffee shop. They offer you $400 for your first week, with a guaranteed raise of $25 every single week. So, week two you make $425, week three you make $450, and so on. It's a steady, predictable increase.
The second offer is from a tech startup, helping them with a new app launch. They also offer $400 for the first week. But instead of a fixed raise, they offer a 5% raise each week. So, week two you'd make $420, and week three you'd make $441. The raise amount itself is getting bigger each time.
Which job pays more by the end of the summer? To answer that, you need to understand the two different types of growth at play. We're going to break down these patterns, called arithmetic and geometric sequences, so you can predict future values for any scenario like this.
Concept overview
flowchart TD
A[Start with a sequence of numbers] --> B{How does it change term-to-term?};
B --> C[Add a constant value, d?];
C -- Yes --> D[Arithmetic Sequence];
D --> E[Use formula: a_n = a_1 + d(n-1)];
B --> F[Multiply by a constant value, r?];
F -- Yes --> G[Geometric Sequence];
G --> H[Use formula: g_n = g_1 * r^(n-1)];
B -- Neither --> I[Not arithmetic or geometric];
Core explanation
Hey everyone, it's Saavi. Today we're diving into the building blocks of many mathematical patterns: sequences. Think of them as predictable, ordered lists of numbers.
What is a Sequence?
A sequence is just a list of numbers that follow a specific rule. For example: 3, 6, 9, 12... is a sequence. Each number in the list is called a term.
In AP Precalculus, we treat sequences as functions. The input is the term's position (1st, 2nd, 3rd, etc.), which is always a whole number. The output is the actual value of the term.
This is why, when we graph a sequence, we get a set of distinct points, not a smooth, connected curve. You can have a 1st term and a 2nd term, but you can't have a 1.5th term, just like you can't have the 1.5th person in line at a movie theater.
Arithmetic Sequences: The Steady Climb
An arithmetic sequence is one where you add the same exact number to get from one term to the next. This constant number is called the common difference, denoted by d.
Consider the sequence: 5, 9, 13, 17, 21...
To get from 5 to 9, you add 4. To get from 9 to 13, you add 4. To get from 13 to 17, you add 4.
The common difference, d, is 4. Because you're adding the same amount each time, this is a "constant rate of change." Sound familiar? It should! It's the same idea as the slope of a line. An arithmetic sequence, when graphed, will always form a set of points that lie on a straight line.
Finding the Formula
How can we find the value of, say, the 100th term without writing everything out? We need a general formula. Let's call our first term a_1.
a_1 = 5a_2 = 5 + 4a_3 = 5 + 4 + 4 = 5 + 2 * 4a_4 = 5 + 4 + 4 + 4 = 5 + 3 * 4
Do you see the pattern? For the nth term, we start with a_1 and add the common difference d exactly (n-1) times.
This gives us our first key formula:
a_n = a_1 + d(n-1)
Sometimes, problems give you a "zeroth" term, a_0, which is the value before the sequence officially starts. Think of it like a y-intercept. In that case, the formula is a bit simpler: a_n = a_0 + dn.
The most flexible formula, which the College Board provides, lets you start from any term a_k:
a_n = a_k + d(n-k)
This is powerful. If you know the 5th term and want the 20th, you don't have to find the 1st term first.
Geometric Sequences: The Exponential Takeoff
A geometric sequence is one where you multiply by the same number to get from one term to the next. This number is called the common ratio, denoted by r.
Consider the sequence: 3, 6, 12, 24, 48...
To get from 3 to 6, you multiply by 2. To get from 6 to 12, you multiply by 2. To get from 12 to 24, you multiply by 2.
The common ratio, r, is 2. This is a "constant proportional change." Each new term is a constant multiple of the one before it.
Finding the Formula
Let's build the formula for a geometric sequence, starting with the first term, g_1.
g_1 = 3g_2 = 3 * 2g_3 = 3 * 2 * 2 = 3 * 2^2g_4 = 3 * 2 * 2 * 2 = 3 * 2^3
The pattern here involves exponents. For the nth term, we start with g_1 and multiply by the common ratio r exactly (n-1) times.
This gives us our formula:
g_n = g_1 * r^(n-1)
Just like with arithmetic sequences, you might be given a "zeroth" term, g_0. This is the initial value before the first multiplication happens. The formula then becomes g_n = g_0 * r^n. This should remind you of the standard exponential function y = ab^x.
And the most flexible version lets you start from any term g_k:
g_n = g_k * r^(n-k)
Growth Spurt: Arithmetic vs. Geometric
Let's go back to the summer job offers.
- Coffee Shop (Arithmetic)$400, $425, $450, $475... (adds $25 each time)
- Startup (Geometric)$400, $420, $441, $463.05... (multiplies by 1.05 each time)
In the first few weeks, the coffee shop job pays more. But watch what happens. The amount of the raise at the startup keeps getting bigger ($20, then $21, then $22.05...). This is the key difference.
- Increasing arithmetic sequencesgo up by the same amount each step. They grow steadily.
- Increasing geometric sequencesgo up by a larger and larger amount each step. They start slow and then explode upwards.
For the AP Exam, you need to be able to look at a scenario, a list of numbers, or a graph of points and immediately identify whether the underlying pattern is one of steady addition (arithmetic) or explosive multiplication (geometric).
Worked examples
Let's walk through a few problems together. The key is to first identify the pattern, then apply the correct formula.
Planning for a Raise
Priya starts a new job as a graphic designer in Dallas with a starting salary of $62,000. Her company guarantees her a raise of $3,000 every year she works there. What will her salary be in her 8th year at the company?
Step 1: Identify the sequence type. The salary increases by a fixed amount ($3,000) each year. This is a constant addition, so we're dealing with an arithmetic sequence.
Step 2: Identify the key values.
- The first term,
a_1, is the starting salary: $62,000. - The common difference,
d, is the annual raise: $3,000. - We want to find the salary in the 8th year, so
n = 8.
Step 3: Choose and apply the formula.
The most straightforward formula here is a_n = a_1 + d(n-1).
Let's plug in our values:
a_8 = 62000 + 3000(8 - 1)
a_8 = 62000 + 3000(7)
a_8 = 62000 + 21000
a_8 = 83000
A Growing Town
The population of a small town in Oregon is 8,500 people in the year 2020. A city planner estimates the population will grow by 2.5% each year. What is the projected population for the year 2030?
Step 1: Identify the sequence type. The population grows by a percentage, which means we are multiplying by a constant factor each year. This is a geometric sequence.
Step 2: Identify the key values.
- The initial population is 8,500. It's helpful to think of 2020 as our "year 0". So,
g_0 = 8500. - The growth is 2.5% per year. This is a classic trap. The common ratio
ris NOT 0.025. The population keeps 100% of its value and adds 2.5%. So,r = 1 + 0.025 = 1.025. - We want the population in 2030, which is 10 years after 2020. So,
n = 10.
Step 3: Choose and apply the formula.
Since we started with g_0, the formula g_n = g_0 * r^n is the best fit.
Plug in the values:
g_10 = 8500 * (1.025)^10
Now, we use a calculator:
g_10 ≈ 8500 * (1.28008)
g_10 ≈ 10880.72
Since we can't have a fraction of a person, we'll round to the nearest whole number.
Finding the Rule from Two Terms
A sequence has a 4th term of 11 and a 9th term of 26. Find a formula for the nth term, a_n.
Step 1: Test for the sequence type.
Let's assume it's arithmetic first, as that's simpler to check. If it is, there's a common difference d. To get from the 4th term to the 9th term, we have to add d five times (9 - 4 = 5).
The total change in value is 26 - 11 = 15.
So, 5d = 15, which means d = 3.
Since we found a constant difference, it is indeed an arithmetic sequence.
Step 2: Choose and apply the formula.
We don't know a_1, so let's use the more flexible formula: a_n = a_k + d(n-k). We can use either of the points we know. Let's use k=4, so a_k = 11.
a_n = a_4 + d(n - 4)
a_n = 11 + 3(n - 4)
Step 3: Simplify the formula.
a_n = 11 + 3n - 12
a_n = 3n - 1
You can quickly check this:
For n=4: a_4 = 3(4) - 1 = 12 - 1 = 11. Correct.
For n=9: a_9 = 3(9) - 1 = 27 - 1 = 26. Correct.
Try it yourself
Ready to try a couple on your own? Remember to identify the sequence type first.
-
Carlos is training for a marathon. On his first day of training, he runs 3 miles. He plans to increase his distance by 1.5 miles each week. How many miles will he run on his 7th week of training? Is this pattern arithmetic or geometric?
Hint: Is the distance changing by a fixed amount or a fixed multiplier? Use that to choose your formula.
-
A rare baseball card is purchased for $200. A collector believes it will appreciate (increase in value) by 15% each year. What would be the formula for the card's value,
V_n, afternyears? What is its estimated value after 4 years?Hint: Be careful when calculating the common ratio
rfor a percentage increase. Remember, the card keeps its old value and adds new value.
Practice — 8 questions
In simple terms, this topic is about two key patterns: adding the same amount each time (arithmetic) or multiplying by the same amount each time (geometric).
- 2.1.A: Express arithmetic sequences found in mathematical and contextual scenarios as functions of the whole numbers.
- 2.1.B: Express geometric sequences found in mathematical and contextual scenarios as functions of the whole numbers.
- 2.1.A.1
- A sequence is a function from the whole numbers to the real numbers. Consequently, the graph of a sequence consists of discrete points instead of a curve.
- 2.1.A.2
- Successive terms in an arithmetic sequence have a common difference, or constant rate of change.
- 2.1.A.3
- The general term of an arithmetic sequence with a common difference d is denoted by a_n and is given by a_n = a_0 + dn, where a_0 is the initial value, or by a_n = a_k + d(n − k), where a_k is the kth term of the sequence.
- 2.1.B.1
- Successive terms in a geometric sequence have a common ratio, or constant proportional change.
- 2.1.B.2
- The general term of a geometric sequence with a common ratio r is denoted by g_n and is given by g_n = g_0 r^n, where g_0 is the initial value, or by g_n = g_k r^{(n-k)}, where g_k is the kth term of the sequence.
- 2.1.B.3
- Increasing arithmetic sequences increase equally with each step, whereas increasing geometric sequences increase by a larger amount with each successive step.
flowchart TD
A[Start with a sequence of numbers] --> B{How does it change term-to-term?};
B --> C[Add a constant value, d?];
C -- Yes --> D[Arithmetic Sequence];
D --> E[Use formula: a_n = a_1 + d(n-1)];
B --> F[Multiply by a constant value, r?];
F -- Yes --> G[Geometric Sequence];
G --> H[Use formula: g_n = g_1 * r^(n-1)];
B -- Neither --> I[Not arithmetic or geometric];
Read what Saavi narrates
Hey everyone, it's Saavi from Shrutam. Let's talk about patterns.
Imagine you get two different summer job offers. One pays you four hundred dollars for the first week, and gives you a twenty-five dollar raise every single week. That's a steady, predictable increase. The second job also starts at four hundred dollars, but offers a five percent raise each week. At first, that five percent is less than twenty-five dollars, but the raise itself gets bigger each time.
Which job is better? To figure that out, you need to understand these two types of patterns.
The first is called an arithmetic sequence, where you add a fixed amount each time. The second is a geometric sequence, where you multiply by a fixed amount. Today, we're going to master both so you can predict the future... at least, when it comes to numbers.
Let's look at an example you might see. A town's population is 8,500 people. A planner predicts it will grow by two-point-five percent each year. What's the population in ten years?
First, we know this is geometric because it's growing by a percentage. We're multiplying. The initial population, which we can call g-zero, is 8,500. The number of years, n, is 10.
Now for the tricky part... the common ratio, r. A lot of students will just use zero-point-zero-two-five. But that would mean the town's population shrinks to almost nothing! You have to keep the original one hundred percent of the people, and add two-point-five percent. So, the ratio r is one plus zero-point-zero-two-five, which is one-point-zero-two-five.
Our formula is g-sub-n equals g-sub-zero times r to the power of n.
So, g-sub-ten equals 8,500 times one-point-zero-two-five to the power of ten.
You'd use your calculator for this, and you get about ten thousand, eight hundred eighty-one people.
One of the most common mistakes I see every year is mixing up the starting term. Is it the first term, a-one, or the initial value, a-zero? If the problem says "starting salary is sixty thousand dollars," that's your first term, a-one. If it says "a population of 500 grows," that 500 is the value at time zero, before any growth has happened. So that's a-zero. Paying close attention to the wording of the problem will tell you which formula to use, `a sub n equals a sub one plus d times n minus one` or `a sub n equals a sub zero plus d n`.
These patterns are everywhere once you start looking for them. Keep practicing identifying them, and you'll be in great shape. You've got this.
The formula `a_n = a_1 + d(n-1)` works because you apply the difference `d` starting from the *second* term. There are `n-1` steps between the 1st and nth term. Using `n` overcounts the steps.
Always ask yourself, "How many steps are there from term 1 to term n?" The answer is `n-1`. This will help you remember the correct factor for both arithmetic and geometric formulas starting from the first term.
Multiplying by 0.04 would mean the value shrinks to just 4% of its previous size. You need to retain the original 100% and *add* the new 4%.
For a growth rate of `p` percent, the common ratio is `r = 1 + (p/100)`. For a 4% growth, `r = 1 + 0.04 = 1.04`. For decay, you subtract.
Applying an addition-based formula to a multiplication-based pattern (or vice-versa) will produce a completely incorrect result.
Remember: **A**rithmetic is for **A**dding. **G**eometric is for... well, it's the other one (multiplying). Connect the "rate of change" in arithmetic sequences to linear functions (`y=mx+b`) and the "proportional change" in geometric sequences to exponential functions (`y=ab^x`).
You're mixing up the core operations. Arithmetic is about differences, geometric is about ratios.
To find `d`, take any term and subtract the previous term (`d = a_n - a_{n-1}`). To find `r`, take any term and divide by the previous term (`r = g_n / g_{n-1}`).
A sequence is a function of the whole numbers. Its domain is `{1, 2, 3, ...}`. A solid line implies the function is defined for all real numbers in between (like 2.5), which is not true for a sequence.
Always graph a sequence as a set of discrete points (a scatter plot).
It's not wrong, but it's an unnecessary and time-consuming extra step.
Use the general formula `a_n = a_k + d(n-k)`. Once you find `d`, you can write the explicit formula immediately using one of the given points, without ever solving for `a_1`. This is faster and less prone to calculation errors.