Could you please prove the following equation from Arithmetic Progressions?

If a_1, ... , a_n are in AP where, a_i > 0 for all i, then show that
1/(sqrt(a_1)+sqrt(a_2))+1/(sqrt(a_2)+sqrt(a_3))+....+1/(sqrt(a_n-1)+sqrt(a_n))= (n-1)/(sqrt(a_1)+sqrt(a_n)).

1 Answer
Cesareo R.
Feb 25, 2018

See below.

Explanation:

If the sequence a_k is an AP then

a_(k+1) = a_k + Delta so

1/(sqrt(a_k)+sqrt(a_(k+1))) = (sqrt(a_(k+1))-sqrt(a_k))/(a_(k+1)-a_k) = (sqrt(a_(k+1))-sqrt(a_k))/Delta

and then

sum_(k=1)^(n-1)1/(sqrt(a_k)+sqrt(a_(k+1))) =1/Delta (sum_(k=1)^(n-1) sqrt(a_(k+1))-sum_(k=1)^(n-1) sqrt(a_k)) = 1/Delta(sqrt(a_n)-sqrt(a_1))

but

1/Delta(sqrt(a_n)-sqrt(a_1)) = 1/Delta(sqrt(a_n)-sqrt(a_1))((sqrt(a_n)+sqrt(a_1))/(sqrt(a_n)+sqrt(a_1))) = 1/Delta(a_n-a_1)/(sqrt(a_n)+sqrt(a_1)) = 1/Delta ((n-1)Delta)/(sqrt(a_n)+sqrt(a_1)) = (n-1)/(sqrt(a_n)+sqrt(a_1))

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