Dear Chandrakesh Shukla

given

            (aⁿ +bⁿ)/(a^n-1 +b^n-1) =(a+b)/2

      2(aⁿ +bⁿ) = (aⁿ +bⁿ) + ab^n-1 + ba^n-1

   (aⁿ +bⁿ) = ab^n-1 + ba^n-1

  a^n-1 (a-b) + b^n-1 ( b-a)=0

  (a^n-1 -b^n-1)(a-b)=0

   (a^n-1 +b^n-1)=0

 (a/b)^n-1 =1

   n-1 =0

  n=1

Please feel free to post as many doubts on our discussion forum as you can.
If you find any question Difficult to understand - post it here and we will get you
the answer and detailed  solution very  quickly.

 We are all IITians and here to help you in your IIT JEE preparation.

All the best.
 
Regards,
Askiitians Experts
Badiuddin

Dear Chandrakesh

 let the series has n term and  the first term is =a 

and comon difference is =d

now find the rth term from the biginning = a+(r-1)d

as we know  rth term from the end  = (n-r+1)th term from the bigining

 so rth term from the end = a+(n-r+1-1)d

                                    =a+(n-r)d

sum of rth term from the beginning and end is  = a+(r-1)d +a+(n-r)d

                                                                  = 2a +(n-1)d

 this term is independent of r so it is a constant

Please feel free to post as many doubts on our discussion forum as you can.
If you find any question Difficult to understand - post it here and we will get you
the answer and detailed solution very quickly.

 We are all IITians and here to help you in your IIT JEE preparation.

 All the best.
 
Regards,
Askiitians Experts
Badiuddin