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Hey there! Today we're going to explore the wonderful world of number patterns. You'll see how numbers can jump and grow in perfectly predictable ways!
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 find the next terms by figuring out the pattern.
An arithmetic sequence is like climbing a staircase. To get from one term to the next, we always add (or subtract) the exact same number, which we call the 'common difference'.
A geometric sequence grows much faster, like a multiplying cell or a spreading rumor. To get to the next term, we multiply by a fixed number called the 'common ratio'.
Identify if the sequence 3, 9, 27, 81, ... is arithmetic or geometric, and find the next term.
- Look at the first two terms: 3 and 9. Let's check for a common difference. 9 - 3 = 6.
- Check the next pair: 27 - 9 = 18. Since 6 is not equal to 18, this sequence is NOT arithmetic.
- Now, let's check for a common ratio by dividing consecutive terms: 9 / 3 = 3.
- Check the next pair: 27 / 9 = 3, and 81 / 27 = 3. Since the ratio is always 3, this is a geometric sequence with a common ratio of 3.
- To find the next term, multiply the last term (81) by the common ratio (3): 81 * 3 = 243.
