Directions : Select the correct alternative from the given choices. There are n terms in an arithmetic progression. The n terms of the arithmetic progression are now distributed into eight sub-series S1, S2 ……and S8 as follows. The 1st, 9th, 17th terms and so on go into S1; the 2nd, 10th, 18th terms and so on go into S2; the 3rd, 11th, 19th terms and so on go into S3, and so on for S4 till S8. If for exactly three of the eight sub-series, the average of the sub-series is a term of the same sub-series, how many of the following values can n assume? (i)43 (ii)53 (iii)51 (iv)69 (v)77

Distributing an arithmetic progression into eight sub-series means each sub-series is itself an arithmetic progression with the same common difference. The average of a finite arithmetic progression is a term of that series only when the series has an odd number of terms. Exactly three of the eight sub-series must have an odd number of terms, which occurs when the total number of terms n leaves a remainder of 3 upon division by 8. Among the given options, dividing 43 by 8 gives a remainder of 3, and dividing 51 by 8 also gives a remainder of 3, while the others do not. Therefore, exactly 2 of the given values can be assumed by n.

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