A sequence of numbers `A_nn=1,2,3` is defined as follows : `A_1=1/2` and for each `ngeq2,` `A_n=((2n-3)/(2n))A_(n-1)` , then prove that `sum_(k=1)^n A_k<1,ngeq1`

Prove that sum_(k=1)^n 1/(k(k+1))=1−1/(n+1) .

Let the sequence a_n be defined as follows: a_1 = 1, a_n = a_(n - 1) + 2 for n ge 2 . Find first five terms and write corresponding series

Prove that sum_(k=1)^(n)(1)/(k(k+1))=1-(1)/(n+1).

If A = [((1)/(2),alpha),(0,(1)/(2))] , prove that sum_(k=1)^(n)"det" (A^(k))=(1)/(3)(1-(1)/(4^(n))).

Find the first five terms of the following sequence . a_1 = 1 , a_2 = 1 , a_n = (a_(n-1))/(a_(n-2) + 3) , n ge 3 , n in N

A sequence a_(1),a_(2),a_(3), . . . is defined by letting a_(1)=3 and a_(k)=7a_(k-1) , for all natural numbers kle2 . Show that a_(n)=3*7^(n-1) for natural numbers.

Find the sequence of the numbers defined by a_(n)={{:(1/n,"when n is odd"),(-1/n,"when n is even"):}

Write the first three terms of the sequence defined by a_1 2,a_(n+1)=(2a_n+3)/(a_n+2) .

Let a_1,a_2,.........a_n be real numbers such that sqrt(a_1)+sqrt(a_2-1)+sqrt(a_3-2)++sqrt(a_n-(n-1))=1/2(a_1+a_2+.......+a_n)-(n(n-3)/4 then find the value of sum_(i=1)^100 a_i