what would be the sum of this series of multiple APs What would be the summation of the following series which effectively is a combination of a kind of two APs with common differences 2 and 3.
2, 5, 8, 10, 13, 16, 19, 21, 24, 27, 29, 32, 35, 38, 40, 43, 46, 48, 51, 54, 57, 59, 62, 65, 67, ...

As you can see, first 3 terms has a common difference of 3
Then, increased by 2,
Then, 4 terms with common difference of 3
Then, increased by 2,
Then, again 3 terms with common difference 3.
and so on.
So, here the series kind has a kind of a pattern. But I am unable to summarize all of them into one series or summation.
 A: The differences are given by the repeating sequence 3,3,2,3,3,3,2.
These are 7 differences that sum up to 19. The sequence can be split in 7 sequences of difference 19.
2, 5, 8, 10, 13, 16, 19, 21, 24, 27, 
    29, 32, 35, 38, 40, 43, 46, 48, 51, 54, 
    57, 59, 62, 65, 67, ...

= 2, 21, 40, 59, ...
+ 5, 24, 43, 62, ...
...
+ 16, 35, 54, ...
+ 19, 38, 67, ...

can you take it from here?

What is the 1000th element of the series?
We have merged 7 series and so we divide 100 by 7 and get
$$1000=7\cdot 142+6$$
So the first 1000 elements of the merged series contains 142 elements of all 7 series and one additional element from the first 6 series.
The sum of the 1000 elements of the series is the sum the sum of the first 143 Elements of the first 6 series and the sum of the first 142 elements of th 7th series.
The 1000th element of the merged series is the 143th element of the 6th series.
Ihe 6th series is 
$$16, 35, 54, \ldots = 16+9(k-1)$$ and the 143th element of this series is 
$$16+9(143-1)=1294$$
which is the 1000th element of the merged series, too
