ISEE Upper Level

Patterns & Sequences

Q: What does "nth term" even mean?

The letter `n` is just a placeholder for the position in line. The "1st term" is the first number, the "2nd term" is the second number. The "nth term" is just a mathy way of saying "a formula that works for ANY spot in the line!"

Arithmetic and geometric sequences, pattern recognition, and nth-term formulas — classified under algebra because the core skill is generalizing a rule, even though SSAT places it under Number Concepts

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Think of mathematical patterns like a secret code in your favorite video game. Once you figure out the rule, you know exactly what happens next! 🎮 In math, we call these lists of numbers "sequences." There are two main types of secret codes you'll need to crack on the ISEE.

The first is called an "arithmetic sequence." This is when you add or subtract the exact same number every time. Imagine building a Lego tower and adding exactly 3 blocks every minute. Your tower goes from 3 blocks, to 6, to 9, to 12. Easy, right? 🧱

The second type is a "geometric sequence." This is when you multiply or divide by the same number each time. Think of a classic zombie movie where the number of zombies doubles every hour: 1, 2, 4, 8, 16! 🧟‍♂️ Things get big really fast!

On the ISEE, the Quantitative Reasoning section might ask you to compare two patterns to see which one is bigger. The Mathematics Achievement section might ask you to find the "nth term," which is a fancy way of saying "what is the 50th number in this list?" Don't worry, you won't have to count all the way to 50. We have cool tricks to jump straight to the answer. Let's crack the code! 🕵️‍♂️

Key Formula

To find any number in an adding/subtracting pattern (arithmetic sequence), use this formula:

a_{n} = a_{1} + (n - 1) d

. Here,

a_{1}

is the first number,

d

is the difference (what you add each time), and

n

is the spot in the line you want to find!

Practice Questions

3 practice questions for ISEE Upper Level

Q1 Hard

The first five terms of an arithmetic sequence of numbers are shown.

4, 1, - 2, - 5, - 8

Which expression represents the

n

th term of this sequence?

- 3 n + 7

- 3 n + 4

n - 3

4 n - 3

Show Solution

The values decrease by $3$ each time, so the sequence is linear with a common difference (slope) of $- 3$ . This means the expression is in the form $- 3 n + b$ . The first term of the sequence (where $n = 1$ ) is $4$ , so substituting these values yields $4 = - 3 (1) + b$ , which gives $b = 7$ . The expression representing the $n$ th term is $- 3 n + 7$ .

Answer: A

Q2 Hard

Marcus is building a tower out of blocks. The first level requires

8

blocks, the second level requires

11

blocks, the third level requires

14

blocks, and the fourth level requires

17

blocks. If this pattern continues, which expression represents the number of blocks required for the

n

th level?

n + 3

3 n + 5

3 n + 8

8 n - 3

Show Solution

The number of blocks in each level forms an arithmetic sequence: 8, 11, 14, 17. The number of blocks increases by $3$ for each subsequent level, so the sequence is linear with a slope of $3$ , meaning it is in the form $3 n + b$ . For the first level ( $n = 1$ ), there are $8$ blocks. Substituting these values gives $8 = 3 (1) + b$ , so $b = 5$ . The expression for the $n$ th level is $3 n + 5$ .

Answer: B

Q3 Hard

The first five terms of a sequence of numbers are shown.

2, 5, 10, 17, 26

Which expression represents the

n

th term of this sequence?

2 n + 3

3 n - 1

n^{2} + 1

n^{2} + n

Show Solution

Notice that the differences between consecutive terms are 3, 5, 7, 9. Since the difference is not constant, the sequence is not linear. Look at the relationship between the term number $n$ and the term value. For $n = 1$ , the value is $2$ . For $n = 2$ , the value is $5$ . For $n = 3$ , the value is $10$ . Each term is exactly $1$ more than a perfect square ( $1^{2} + 1 = 2$ , $2^{2} + 1 = 5$ , $3^{2} + 1 = 10$ , etc.). Therefore, the expression for the $n$ th term is $n^{2} + 1$ .

Answer: C

Tips & Strategies

Plug in the answers! If an ISEE question asks for the $n$ th term formula, you don't actually have to build it from scratch. Just pick $n = 1$ (the first number in the pattern) and plug it into the answer choices. Whichever choice gives you the correct first number is usually the winner!
For Quantitative Comparison questions, don't just guess if a pattern looks like it will get big. Actually write out the next few terms or use the formula to find the exact number. The ISEE loves to make Column B almost the right answer to trick you.
Sometimes patterns use fractions! If you see a sequence like $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}$ , treat the top numbers (numerators) as one pattern and the bottom numbers (denominators) as a separate pattern.

Common Mistakes

Watch out for stopping after the first jump! A common mistake is assuming a pattern is adding when it's actually multiplying. Always test your rule on the third and fourth numbers in the sequence.
Don't forget that $n$ stands for the position of the number, not the number itself! If you want the 5th number in the sequence, you plug in $n = 5$ , not the value of the 4th number.

Frequently Asked Questions

What if I can't figure out the pattern rule at all?

Don't panic! On the ISEE, there is no penalty for guessing. If you're stuck, look at the answer choices. Sometimes working backward by plugging the answer choices into the sequence helps you see the hidden rule.

Do I need to memorize complicated formulas for the ISEE?

Not really! While the official formula is helpful, you can often just find the "jump" amount. If the jump is $+ 4$ , the rule starts with $4 n$ . Then just adjust it to match the first number by adding or subtracting.

What does "nth term" even mean?

The letter $n$ is just a placeholder for the position in line. The "1st term" is the first number, the "2nd term" is the second number. The "nth term" is just a mathy way of saying "a formula that works for ANY spot in the line!"

Why are there Column A and Column B questions?

That's the Quantitative Reasoning section! The ISEE uses these to test your logical thinking. Instead of just calculating an answer, you have to compare two values. Remember, the choices are always (A) Column A is greater, (B) Column B is greater, (C) Equal, or (D) Cannot be determined.

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