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For example, suppose I am given the following:

$$ a_n = \frac{\pi}{8} + (-1)^n \frac{n \pi}{4}$$

$$ b_n = \frac{\pi}{4} + (-1)^n n \frac{\pi}{2}$$

Now, from these two sequences, how do I find the sequences which contains the intersection terms of both(i.e: a new sequence in which terms are common to both a_n and b_n?)

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  • $\begingroup$ Do you mean $n$ instead of $k$ for $b_n$? $\endgroup$
    – daruma
    Jan 21, 2021 at 18:43
  • $\begingroup$ You could start by looking at the first few terms of each sequence. That should convince you that there can only be a finite number of intersections and possibly none $\endgroup$
    – Henry
    Jan 21, 2021 at 18:45
  • $\begingroup$ @daruma my bad , it waws typo $\endgroup$ Jan 21, 2021 at 21:15

1 Answer 1

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Take a common denominator of 8 everywhere and remove the common factor of $\frac{\pi}{8}$.

The first sequence is $1 + (-1)^{n}2n$ and is always odd.

The second sequence is $2 + (-1)^{n}4n$ and is always even.

The two never overlap. Even numbers cannot be odd and vice versa.

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  • $\begingroup$ Fixed the typo @Open problem $\endgroup$ Jan 21, 2021 at 21:15

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