Find 12th Term in Simple Sequence: 1,4,9,16...

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In summary, the conversation discusses the sequence of numbers 1, 4, 9, 16... and the 12th term in the sequence. The textbook states that the 12th term is 144 since each term is the square of its value. However, there are alternate algorithms that can generate the same sequence, such as adding 3, 5, 7 to each term. The conversation also mentions that there are uncountably many integer sequences that can continue the given partial sequence. The discussion concludes that the sum of the first n odd numbers is n^2, which is why the sequence can also be generated by adding consecutive odd numbers.
  • #1
ila
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If there is a sequence of numbers 1,4,9,16... what is the 12th term in the sequence (where 1 is the first term). The textbook says that it is 144 since each term of the sequence is the square of the term value. I find (and have always found) this to be confusing, why can't an alternate algorithm generate those few numbers, namely add 3 for the first term, five to the second term, 7 to the second term. Now there there appears to me an obvious pattern, the amount that one is adding to each term is increasing by 2 in the number of terms. I still haven't figured out an expression for the nth term for this alternative though. The general point though is this, with these sequences if we have given only a few numbers why can't we propose many different algorithms and then suggest that that is the correct generating process?
 
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  • #2
1. There are uncountably many integer sequences that continue the partial sequence you were given.
2. Your sequence happens to be the same as your teacher's: the squares are the sum of consecutive odd numbers!
 
  • #3
You are certainly correct that there are other algorithms that would generate this same sequence for the first few terms. So strictly speaking the next number could be anything. Ironically though if you continue you the sequence with your pattern you will arrive at exactly the same sequence the book does, namely the perfect squares. And it is not too terribly difficult to prove this fact, that the sum of the first n odd numbers is n^2.
 
  • #4
d_leet said:
Ironically though if you continue you the sequence with your pattern you will arrive at exactly the same sequence the book does, namely the perfect squares. And it is not too terribly difficult to prove this fact, that the sum of the first n odd numbers is n^2.
I had a sinking suspicion that that may have been the case.
 
  • #5
All you have to do is see that n^2 = ((n-1) +1)^2 = (n-1)^2+2(n-1)+1 = (n-1)^2 + (2n-1). So that, 1, 1+3=4, 4+5=9, 9+7= 16. So that each term increases by the next odd number, and is a square.

Thus the nth term is [tex] \sum_1^n (2j-1)=n^2. [/tex]
 
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Related to Find 12th Term in Simple Sequence: 1,4,9,16...

1. What is a simple sequence and how is it different from other types of sequences?

A simple sequence is a mathematical pattern where each term is generated by adding a constant value to the previous term. This is different from other types of sequences, such as geometric or arithmetic sequences, where the pattern involves multiplying or adding a variable value to the previous term.

2. How can I find the 12th term in a simple sequence?

To find the 12th term in a simple sequence, we need to first identify the constant difference between each term. In this case, the constant difference is 3 (1 + 3 = 4, 4 + 3 = 9, etc.). Then, we can use the formula tn = t1 + (n-1)d, where tn is the nth term, t1 is the first term, and d is the constant difference. Plugging in the values, we get t12 = 1 + (12-1)3 = 1 + 33 = 34. Therefore, the 12th term in this sequence is 34.

3. Can a simple sequence have a negative difference between terms?

Yes, a simple sequence can have a negative difference between terms. In this case, the pattern would involve subtracting a constant value from the previous term to generate the next term. For example, the sequence -2, -5, -8, -11, ... is a simple sequence with a constant difference of -3.

4. How can I use a simple sequence in real-world applications?

Simple sequences can be used to model real-world situations where there is a constant rate of change. For example, a simple sequence could represent the growth of a population over time, the increase in stock prices, or the depreciation of a car's value.

5. Is it possible for a simple sequence to have a variable difference between terms?

No, by definition, a simple sequence has a constant difference between terms. If the difference between terms changes, then it would be considered a different type of sequence. However, a simple sequence can have a variable initial term, meaning the pattern starts with a different number and then follows a constant difference. For example, the sequence 2, 5, 8, 11, ... has a constant difference of 3, but the initial term is not 1 as in the original sequence.

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