I felt this qsn was nice on sequences....I liked it....
Let \left\{t_n \right\} define a sequence of reals, such that sum of every 5 terms is positive, while sum of every 9 terms is negative.
Find the maximum number of terms in the sequence.
1Another old question ............ no takers ?
t1 + t2 + t3 + t4 + t5 < 0
.............................
t9 + t10 + t11 + t12 + t13 < 0
Adding these row - wise , we find that the total sum is negative , however , adding them column - wise yields a positive sum altogether - Contradiction !
Hence , this sequence is terminated after 13 terms .
1In fact , as a generalisation , we can change " 5 " and " 9 " to two positive integers " p " and " q " such that ,
gcd ( p , q ) = d
and obtain the maximum number of terms as : - " p + q - d - 1 "
In our problem , plug in -
p = 5 , q = 9 , d = 1 .
Straightforward , we get the answer as - 13
