Patterns & Sequences
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! 🕵️♂️
Practice Questions
5 practice questions for ISEE Lower Level
PLANT GROWTH
| Week | Height |
|---|---|
| Week 1 | 4 cm |
| Week 2 | 5 cm |
| Week 3 | 7 cm |
| Week 4 | 10 cm |
| Week 5 | 14 cm |
According to the pattern in the data, what will be the predicted height of the plant at Week 7?
Show Solution
- Find the pattern of the plant's growth each week by finding the difference between the heights.
From Week 1 to Week 2, the plant grows 1 cm (5 - 4 = 1).
From Week 2 to Week 3, it grows 2 cm (7 - 5 = 2).
From Week 3 to Week 4, it grows 3 cm (10 - 7 = 3).
From Week 4 to Week 5, it grows 4 cm (14 - 10 = 4).
The growth amount increases by 1 cm each week.
To find Week 6, add 5 cm to Week 5's height: 14 + 5 = 19 cm.
To find Week 7, add 6 cm to Week 6's height: 19 + 6 = 25 cm.
5, 8, 14, 23, 35, ___
What is the next number in the sequence?
Show Solution
- Find the difference between each pair of consecutive numbers in the sequence:
8 - 5 = 3
14 - 8 = 6
23 - 14 = 9
35 - 23 = 12
The differences are multiples of 3: 3, 6, 9, 12. The next difference should be 15. Add 15 to the last number in the sequence: 35 + 15 = 50.
What is the solution to ?
Show Solution
- Look at the pattern. The sum of the first even numbers is equal to .
1 even number:
2 even numbers:
3 even numbers:
4 even numbers:
The expression is the sum of the first 7 even numbers. Following the pattern, the solution is .
Show Solution
- Write out the sequence of blocks used for each step: 2, 5, 8, 11.
The pattern increases by 3 blocks each time (+3).
- Step 5: 11 + 3 = 14
- Step 6: 14 + 3 = 17
- Step 7: 17 + 3 = 20
- Step 8: 20 + 3 = 23
The eighth step will use 23 blocks.
80, 78, 74, 68, 60, ___
What is the next number in the sequence?
Show Solution
- Find the difference between each pair of consecutive numbers in the sequence:
80 - 78 = 2
78 - 74 = 4
74 - 68 = 6
68 - 60 = 8
The numbers are decreasing, and the amount subtracted increases by 2 each time (-2, -4, -6, -8). The next amount to subtract is 10. Subtract 10 from the last number: 60 - 10 = 50.
Tips & Strategies
- Plug in the answers! If an ISEE question asks for the th term formula, you don't actually have to build it from scratch. Just pick (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 , 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 stands for the position of the number, not the number itself! If you want the 5th number in the sequence, you plug in , 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 , the rule starts with . Then just adjust it to match the first number by adding or subtracting.
What does "nth term" even mean?
The letter 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.