Hey there! Ready to discover the hidden patterns in numbers? Today, we'll explore sequences and see how they grow step-by-step!
A sequence is simply an ordered list of numbers that follows a specific rule. Each number in the list is called a term, and we can predict what comes next by finding the pattern.
In an arithmetic sequence, we add the same number every time to get to the next term. This constant number we add is called the common difference.
In a geometric sequence, instead of adding, we multiply each term by the same number to get the next one. This multiplier is called the common ratio.
✏️ Worked example
Identify if the sequence 4, 12, 36, 108, ... is arithmetic or geometric, and find the next term.
Look at the first two terms: 4 and 12. Let's test for an arithmetic pattern. To get from 4 to 12, we would add 8.
Test this with the next terms: 12 + 8 = 20, but our next term is 36. So, it is not an arithmetic sequence.
Now, let's test for a geometric pattern. To get from 4 to 12, we multiply by 3 (since 12 ÷ 4 = 3).
Check if this multiplier works for the next terms: 12 × 3 = 36, and 36 × 3 = 108. Yes, it works! The common ratio is 3.
To find the next term, multiply the last term by the common ratio: 108 × 3 = 324.
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