How would you derive a new series from two existing?

[Johulus]

How could you derive a new arithmetical series from the two existing ones?

For example:

5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,95,101,....
3,8,13,18,23,28,33,38,43,48,53,58,63,68,73,78,83,88,93,....

We can clearly see that the two series coincide for 23,53,83,....

But Could we somehow calculate this without writing out the members of these series:

For the first series \(\displaystyle a_n=5+(n-1)6 \) and for the second \(\displaystyle a_n=3+(n-1)5 \) . Could we somehow get this new series from only this information.

So, for now I think I can get d for the new series it should be 5*6=30, but how could I get the first member of the series only using the formulas of these two series?

 

[Johulus]

Depends on what you're trying to do...combine in what manner?
Could easily be: 8, 19, 30, 41....

So that a new series consists of numbers that are mutual to both initial series.

Could it be done without writing out the numbers from each series and observing? Like using the formula for the general term of both series to come up with the series whose members are the numbers that are the same for both series(mutual).

For example: 2,5,8,11,....
2,4,6,8,10,....
Here we have two series. Their mutual terms are 2 and 8. Could I now somehow use the formula for genral term of each series to determine the new series with the mentioned property without writing out the numbers from these series.

 

[Ishuda]

So that a new series consists of numbers that are mutual to both initial series.

Could it be done without writing out the numbers from each series and observing? Like using the formula for the general term of both series to come up with the series whose members are the numbers that are the same for both series(mutual).

For example: 2,5,8,11,....
2,4,6,8,10,....
Here we have two series. Their mutual terms are 2 and 8. Could I now somehow use the formula for genral term of each series to determine the new series with the mentioned property without writing out the numbers from these series.

As you did in the first example. The difference for the new series will be the least common multiple of the differences for the two initial series (in this case 6 is the lcm of 2 and 3) combined with a common term of both series (in this case 2) to get the new series
6n-4; n=1, 2, 3, ... : 2, 8, 14, ...

 

[lookagain]

How could you derive a new arithmetical > > series < < from the two existing ones?

For example:

5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,95,101,....
3,8,13,18,23,28,33,38,43,48,53,58,63,68,73,78,83,88,93,....

Those are sequences. (They're just lists of numbers; you're not adding the numbers up.)

You could derive a new arithmetic sequence by subtracting the second one from the first (corresponding terms):

2, 3, 4, ...