here,

where n = 1,2,3,....

so, for 6th term n = 6

nth term of given sequence is n(n+1) so the 5th term is 5(5+1)=30

Terms of given sequence can be written as

so the required term is 33/32

for real n, imaginary part = 0

so 2n - 8=0

which gives, n = 4

the series obtained will not follow rules of AP, GP and HP

As a,4,b are in AP so, ................(i)

Also a,2,b are in GP so, ab = 4...................(ii)

from (i) and (ii)

hence a, 1 , b are in HP

here,

sum of n number of A.M's between two numbers a and b is given by

so,

therefore sum of 50 A.M's between a and b is,