What is a spoken sequence?
I was in math class and one of the examples was: 29787, 13, 2, 1, 1, 1, ...
What does this mean? I learned that I never have to find the general term, but how do I find the next?
Do you mean a look and say sequence?
OH! Yes! I couldn't seem to find the word! How would I solve the example?
A spoken sequence refers to a sequence of numbers or other elements arranged in a specific order. In the example you provided, the sequence is: 29787, 13, 2, 1, 1, 1, ...
To find the next term in a spoken sequence, there are a few different methods you can use. I'll explain two common approaches: pattern recognition and calculating the differences.
1. Pattern Recognition:
Observe the sequence and look for any patterns or relationships between the terms. Ask yourself questions like: Are the numbers increasing or decreasing? Are there any repeating patterns or cycles? Are there any arithmetic or geometric progressions? Based on these observations, try to identify any rules or formulas that might describe the sequence.
In the given example, we can observe that the sequence appears to be decreasing exponentially. The first term decreases by a factor of approximately 2265 (29787 divided by 13), the second term by a factor of approximately 6.5 (13 divided by 2), and the third term by a factor of 2, and so on. This suggests that the next term would likely be obtained by dividing the previous term by 2.
2. Calculating the Differences:
Another method is to calculate the differences between consecutive terms and look for any patterns or relationships in the differences. You can use these patterns to make predictions about the next term.
In the given example, let's calculate the differences between consecutive terms:
13 - 29787 = -29774
2 - 13 = -11
1 - 2 = -1
1 - 1 = 0
1 - 1 = 0
Observe that the differences are decreasing by 1 each time, until they reach 0. This suggests that the next difference will likely be -1. By adding -1 to the last term in the sequence (1), you can predict that the next term will be 1 + (-1) = 0.
It's important to note that finding the next term in a sequence is not always straightforward, and sometimes there may be multiple valid answers or different patterns. However, by using these methods of pattern recognition and calculating differences, you can make educated guesses and refine your understanding of the sequence.