For the recursively defined sequence a1 0 a2 1 an 2 an

For the recursively defined sequence a_1 = 0, a_2 = 1, a_n + 2 = a_n + 1 + 3a_n, a_4 = If in an arithmetic sequence a_3 = 2, a_3 = 8, then a_4 = __ sigma_k = 1^3 (-1)^k+1 1/k + 1 = Tan^-1 (squareroot 3) = __

Solution

a) a3 = a2 + 3a1 = 1 + 0 = 1

a4 = a3 + 3a2 = 1 + 3*1 = 4

a4 = 4

b) a3 = 2,, a5 = 8

2d = 6 ====> d = 3

a9 = a5 + 3*4 = 8 + 12 = 20

a9 = 20

d) Tan^-1 (sqrt(3)) = 60 degree = pi/3