SOLUTION: An arithmetic progression P consists of n terms. From the progression three different progressions P1, P2 and P3 are created such that P1 is obtained by the 1st, 4th, 7th .... term

Question 935076: An arithmetic progression P consists of n terms. From the progression three different progressions P1, P2 and P3 are created such that P1 is obtained by the 1st, 4th, 7th .... terms of P, P2 has the 2nd,5th, 8th .....terms of Pand P3 has 3rd, 6th, 9th ..... terms of P. It is found that of P1, P2 and P3 two progressions have the property that their average is itself a term of the original progression P. Which of the following can be a possible value of n?
(a.) 20
(b.) 26
(c.) 36
(d.) Both 1 and 2

For the above question, if the Common difference between the terms of P1 is 6, what is the common difference of P?
(a.) 2
(b.) 3
(c.) 6
(d.) Cannot be determined
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Let the terms of be , , ... , with , being the common difference between the terms od .

1) The average of terms of an arithmetic progression is term number of that arithmetic progression if is odd.
If terms is the average of two consecutive terms in the middle of that progression (terms number and ), and that is not a term of the arithmetic progression.
If has an odd number of terms, the average of all the terms of will be a term of and a term of .
If has an even number of term its average will be the average of two consecutive terms of that would be terms and of .
In , after comes , , and {{a[h+3]=a[h]+3d}}} . The average of and would be
.
That is a number between and , and is not a term of .
In sum, if has an odd number of terms, the average of its terms is a term of and a term of ; if has an even number of terms, the average of its terms is neither a term of nor a term of .
The same can be said of and .

If has terms, will have terms,
starting with term number of , and ending with term number of . and will also have terms, because starts and ends terms of before , and starts and ends terms of before . In that case, all four arithmetic progressions will have an even number of terms, and will have averages that are not terms of .

If were less, would have less, but and would still have terms. That happens for all that are short of a multiple of , such as
and .
For , has 6 terms, but and have terms. The averages for and are their respective 4th terms, that are terms of .
The same thing happens for , which gives and terms and their averages are their respective 5th terms.

So, the answer is (d.) Both 1 and 2.

2) The common difference for terms of , and is the same. It is the difference between terms and of .
If that is ,
--->--->

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