" 3."a_(n)=2^(n)

Let n in N . If (1+x)^(n)=a_(0)+a_(1)x+a_(x)x^(2)+ . . . .+a_(n) x^(n) and a_(n-3),a_(n-2),a_(n-2),a_(n-1) are in A.P. then:

If a_(1)=3 and a_(n)=2a_(n-1)+5 , find a_(4) .

Let a_(1)+a_(2)+a_(3), . . . ,a_(n-1),a_(n) be an A.P. Statement -1: a_(1)+a_(2)+a_(3)+ . . . +a_(n)=(n)/(2)(a_(1)+a_(n)) Statement -2 a_(k)+a_(n-k+1)=a_(1)+a_(n)" for "k=1,2,3, . . . , n

Let a_(1)+a_(2)+a_(3), . . . ,a_(n-1),a_(n) be an A.P. Statement -1: a_(1)+a_(2)+a_(3)+ . . . +a_(n)=(n)/(2)(a_(1)+a_(n)) Statement -2 a_(k)+a_(n-k+1)=a_(1)+a_(n)" for "k=1,2,3, . . . , n

If a_(1)=1, a_(2)=5 and a_(n+2)=5a^(n+1)-6a_(n), n ge 1 . Show by using mathematical induction that a_(n)=3^(n)-2^(n)

If a_(1)=1, a_(2)=5 and a_(n+2)=5a_(n+1)-6a_(n), n ge 1 . Show by using mathematical induction that a_(n)=3^(n)-2^(n)

A sequence is defined as follows : a_(1)=3, a_(n)=2a_(n-1)+1 , where n gt 1 . Where n gt 1 . Find (a_(n+1))/(a_(n)) for n = 1, 2, 3.

A sequence is defined as follows : a_(1)=3, a_(n)=2a_(n-1)+1 , where n gt 1 . Where n gt 1 . Find (a_(n+1))/(a_(n)) for n = 1, 2, 3.

Write the first five terms of each of the following sequences whose n th terms are: a_(n)=3n+2( ii) a_(n)=(n-2)/(3)a_(n)=3^(n)( iv )a_(n)=(3n-2)/(5)a_(n)=(-1)^(n)*2^(n)( vi) a_(n)=(n(n-2))/(2)a_(n)=n^(2)-n+1 (vii) a_(n)=2n^(2)-3n+1a_(n)=(2n-3)/(6)