Let `a_1, a_2, ..., a_n`. be sequence of real numbers with `a_(n+1) = a_n+ sqrt(1+a_n^2) and a_0 = 0`. Prove that `lim_(n->oo) (a_n/2^(n-1))=2/pi`.

Let a_(1),a_(2),...,a_(n) .be sequence of real numbers with a_(n+1)=a_(n)+sqrt(1+a_(n)^(2)) and a_(0)=0 Prove that lim_(n rarr oo)((a_(n))/(2^(n-1)))=(2)/(pi)

Let a_(1),a_(2),a_(n) be sequence of real numbers with a_(n+1)=a_(n)+sqrt(1+a_(n)^(2)) and a_(0)=1 . Prove that lim_(xtooo)((a_(n))/(2^(n)))=4/(pi)

If a_1, a_2,......,a_n are n distinct odd natural numbers not divisible by any prime greater than 5, then prove that (1)/(a_1)+(1)/(a_2)+…..+(1)/(a_n) lt 2 .

A sequence a_1,a_2,a_3,.a_n of real numbers is such that a_1=0, |a_2|=|a_1+1|,|a_3|=|a_2+1|,gt,|an|=|a_(n-1)+1| . Prove that the arithmetic mean (a_1+a_2+……..+a_n)/n of these numbers cannot be les then - 1/2.

"If "a_1,a_2,a_3,.....,a_n" are in AP, prove that "a_(1)+a_(n)=a_(r)+a_(n-r+1)""

If the sequence {a_n}, satisfies the recurrence, a_(n+1) = 3a_n - 2a_(n-1), n ge2, a_0 = 2, a_1 = 3, then a_2007 is

Let a_1,a_2,a_3 ......, a_n are in A.P. such that a_n = 100 , a_40-a_39=3/5 then 15^(th) term of A.P. from end is

The n^(th) term, an of a sequence of numbers is given by the formula a_n = a_(n – 1) + 2n for n ge 2 and a_1 = 1 . Find an equation expressing an as a polynomial in n. Also find the sum to n terms of the sequence.