An arithmetic progression P consists of terms. From the progression th

1.	An arithmetic progression P consists of 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 P and P3 has the 3rd, 6th, 9th, terms of P. It is found that of P1, P2 and P1 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 (a) and (b)
Answer» Correct option (d) Both (a) and (b) Explanation: The key to this question is what you understand from the statement— ‘for two progressions out of P₁, P₂ and P₃ the average is itself a term of the original progression P.’ For option (a) which tells us that the Progression P has 20 terms, we can see that P₁ would have 7 terms, P₂ would have 7 terms and P₃would have 6 terms. Since, both P₁ and P₂ have an odd number of terms we can see that for P₁ and P₂ their 4th terms (being the middle terms for an AP with 7 terms) would be equal to their average. Since, all the terms of P₁, P₂ and P₃ have been taken out of the original AP P, we can see that for P₁ and P₂ their average itself would be a term of the original progression P. This would not occur for P₃ as P₃, has an even number of terms. Thus, 20 is a correct value for n. Similarly, if we go for n = 26 from the second option we get: P₁, P₂ and P₃ would have 9, 9 and 8 terms, respectively and the same condition would be met here too. For n = 36 from the third option, the three progressions would have 12 terms each and none of them would have an odd number of terms. Thus, option (d) is correct as both options (a) and (b) satisfy the conditions given in the problem

An arithmetic progression P consists of 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 P and P3 has the 3rd, 6th, 9th, terms of P. It is found that of P1, P2 and P1 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 (a) and (b)

Answer»

Correct option (d) Both (a) and (b)

Explanation:

The key to this question is what you understand from the statement— ‘for two progressions out of P₁, P₂ and P₃ the average is itself a term of the original progression P.’ For option (a) which tells us that the Progression P has 20 terms, we can see that P₁ would have 7 terms, P₂ would have 7 terms and P₃would have 6 terms. Since, both P₁ and P₂ have an odd number of terms we can see that for P₁ and P₂ their 4th terms (being the middle terms for an AP with 7 terms) would be equal to their average. Since, all the terms of P₁, P₂ and P₃ have been taken out of the original AP

P, we can see that for P₁ and P₂ their average itself would be a term of the original progression P. This would not occur for P₃ as P₃, has an even number of terms. Thus, 20 is a correct value for n. Similarly, if we go for n = 26 from the second option we get:

P₁, P₂ and P₃ would have 9, 9 and 8 terms, respectively and the same condition would be met here too. For n = 36 from the third option, the three progressions would have 12 terms each and none of them would have an odd number of terms. Thus, option (d) is correct as both options (a) and (b) satisfy the conditions given in the problem

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