Why Do We Need to Predict Terms?
People have been fascinated by number patterns for thousands of years. Ancient mathematicians noticed that certain numbers follow a rule, and if you know the rule, you can figure out any number in the list — even the 100th or the 1,000th! A list of numbers that follows a rule is called a sequence. On the SHSAT, you'll often need to find a term far down the line without writing out every single term. That's exactly what this lesson teaches you.
The big question is: how do you jump straight to a later term without listing every number before it? That's the skill we'll build in this lesson.
Core Ideas Behind Sequences
Before we start predicting later terms, let's nail down a few key ideas. Every sequence question on the SHSAT depends on these building blocks.
Term
Position Number (n)
Common Difference (d)
Pattern Rule (Formula)
Seeing the Pattern
Let's look at a simple sequence and see how the position number connects to each term. In the diagram below, each circle is a term. The number on top is the position (n), and the number inside is the term's value.
Notice how each term grows by exactly 3. That +3 is the common difference. The formula at the bottom is the magic shortcut. You plug in the position number (n), and out comes the term value. No need to count up one by one!
The Formula for Predicting Terms
The most common type of sequence on the SHSAT is the arithmetic sequence (a sequence where you add or subtract the same number each time). Here is the formula you need to know.
Why does the formula use (n − 1) instead of just n? Think about it this way: the first term hasn't been changed at all — you haven't added d yet. The second term has d added once. The third term has d added twice. So the number of times you add d is always one less than the position number.
Recognizing Different Sequence Types
Before you can predict a later term, you need to figure out what kind of sequence you're looking at. Here's a comparison chart and a visual to help you tell them apart.
| Feature | Arithmetic Sequence | Geometric Sequence |
|---|---|---|
| Rule | Add (or subtract) the same number | Multiply (or divide) by the same number |
| Example | 4, 7, 10, 13, 16, … | 3, 6, 12, 24, 48, … |
| Key Number | Common difference d = 3 | Common ratio r = 2 |
| Formula | aₙ = a₁ + (n − 1) × d | aₙ = a₁ × r⁽ⁿ⁻¹⁾ |
| Growth | Steady (straight line on a graph) | Fast (curve on a graph) |
On the SHSAT, always check the differences between terms first. If the differences are the same, it's arithmetic. If the ratios are the same, it's geometric. Once you know the type, you know which formula to use.
Worked Example — Finding the 50th Term
Let's work through a full problem, step by step, just like you would on the SHSAT.
Common Mistakes and How to Avoid Them
Even if you understand the formula, small mistakes can cost you points on test day. Here are the most common errors students make — and how to dodge them.
| Common Mistake | Why It Happens | How to Fix It |
|---|---|---|
| Using n instead of (n − 1) | Students forget that you add d starting from the second term, not the first. | Always check: plug n = 1 into your formula. You should get a₁. |
| Wrong sign on d | If the sequence decreases (like 20, 17, 14, …), d is negative (−3), not positive. | Subtract: second term minus first term. Keep the sign! |
| Confusing arithmetic with geometric | Students use the addition formula when the pattern uses multiplication. | Check differences AND ratios between terms before choosing a formula. |
| Order of operations error | Students add a₁ + (n − 1) first, then multiply by d. | Multiply (n − 1) × d first, then add a₁. Parentheses and multiplication come before addition. |
Connecting to Harder Sequence Questions
The arithmetic sequence formula is the foundation, but harder test questions might twist the idea. Here's a look at what you might see on the most challenging SHSAT questions — and how the same thinking still applies.
| What You Know Now | What Comes Next |
|---|---|
| Find the nth term of an arithmetic sequence | Find the sum of the first n terms (Gauss's trick!) |
| Recognize a constant common difference | Handle sequences where the difference itself changes (quadratic sequences) |
| Use the formula aₙ = a₁ + (n − 1) × d | Write functions like f(x) = mx + b (linear functions are arithmetic sequences!) |
| Identify arithmetic vs. geometric | Work with other patterns: Fibonacci-type, alternating, or rule-based sequences |
Here's a cool connection: the arithmetic sequence formula aₙ = a₁ + (n − 1) × d is really just a linear function in disguise. If you expand it, you get aₙ = d × n + (a₁ − d). That's the same form as y = mx + b. The common difference d is the slope! So every time you find a later term, you're actually doing algebra with linear functions.
Practice Problems
Try these five problems on your own. They start easy and get harder. After each one, check the answer and read the explanation.
Lesson Summary
A sequence is a list of numbers that follows a rule. In an arithmetic sequence, you add the same number (the common difference, d) each time. To predict any later term without listing them all, use the pattern rule formula: aₙ = a₁ + (n − 1) × d. Just plug in the position number (n) and the formula gives you the term.
Always follow these steps: find d by subtracting consecutive terms, identify a₁ (the first term), write the formula, plug in n, and simplify carefully using order of operations. Check your answer by testing a position you already know. Remember, the formula is really a linear function — a powerful connection you'll use again and again in algebra.