Question

let the sequence 1,3,5,7,9_ _ _ to (2n+1) taken. show that sum of alternate terms 1,5,9_ _ _ to the sum of the remaining terms 3,7,11 _ _ _ is in the ratio of (n+1):n.

Answer 1

No space for the note....
A nice problem of AP...

No space for the note in the snap...

You can understand it as...Suppose we have a sequence like 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8

Clearly number of term , n = 8
If we make a squence of alternate terms like
1 , 3 , 5 , 7 and 2 , 4 , 6 , 8 then we must have number of term in each of the squence as half the original squence...see number of term in each sequence = 4 ( half of terms of original sequence..)

....Similarly in the problem given above...we have
2n + 1 terms in the squence...for the alternate squences we must have number of term equal to half the terms in the original squence...

So , number of terms in each squence is ( 2n + 1 ) / 2...