Challenge for math lovers:
Find the general formula for the following sequence:
[tex] 5, 7, 10, 14, 15, 21, 20, 28, 25, 35, 30, 42, ... [/tex]
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Challenge for math lovers:
Find the general formula for the following sequence:
[tex] 5, 7, 10, 14, 15, 21, 20, 28, 25, 35, 30, 42, ... [/tex]
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The general formula of the sequence is:
(using ceiling function)
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Let to be the general term of the sequence. The rule we apply for the terms alternates between odd and even places; when the index is odd, will be multiple of 5, and when the index is even, will be a multiple of 7.
Case 1: When is odd.
If is odd, then we can set for some positive integer It follows that is the positive odd integer. It can be observed that the terms in odd places in the sequence are the multiple of 5. Therefore
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Case 2: When is even.
Same approach. If is even, then we can set for some positive integer It follows that is the positive even integer. In this case, the terms in even places are the multiple of 7 so
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Combining the results, we can get that
The sign becomes negative if is odd and becomes positive if is even. Because the signs are alternating between even and odd, we can just write those as a power of -1;
Note that the general term must be in terms of We know that if is odd and if is even, solving for on each, we have
We need to make the values of on both cases to be equal in order to generalize the value of . To eliminate 1/2 on the case when is odd, we will use the ceiling function. One of the properties of the ceiling function is:
Applying the ceiling function, we get
Hence the general term,
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Tough problem, lol.