Can’t we use the concept of geometric progression to solve this problem?
a(n) can be found out from T^(n-1)*b.
similarly a(m) from T^(m-1)*b.
since we need to find out a(m)+a(m+1)+…a(n).
We can write this sum as :
T^(m-1)*b +T^(m)*b +… +T^(n-1)*b;
which can be further written as [ T^(m-1) + T^(m) + …+T^(n-1) ]b.
Taking T^(m-1) common,we get [ 1+ T+ T^2+ T^3+…+ T^x ] where x=n-m.
using GP we get the sum as [ T^(x+1) -1) ].
We can then further simplify T^(m-1) [ T^(x+1) -1] = T^n -T^(m-1), as x=n-m.
Finally,
[T^n -T^(m-1)]*b we give us the sum required.
I tried this approach but it did not work. Can you help me find out the logical mistake that i am making?