How do I mathematically show that the common terms between the series $3+7+11+....$ and $1+6+11+....$ form an arithmetic progression without actually finding all the individual terms. How does the LCM of the common differences of the given series becomes the common difference of the new series?
My Attempt:
$$
3+(n-1)4=1+(m-1)5\implies 4n-1=5m-4\implies5m-4n=3
$$
But this does not tell me the above statement unless I try all the integer combinations of $m$ and $n$.
Answer
Continuing from your work you can see that the final equation is a simple linear Diphantene equation. Solving it for $m$ (you can also solve it with respect to $n$) you will get $m = 4t + 3$. Plugging it into the coresponding form of the sequence we get that the common terms are given by:
$$1 + 5(m-1) = 1 + 5(4t + 3 - 1) = 20t + 11$$
Hence the common terms make an arithemtic progression and it's given by $c_n = 20n + 11$.
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