Find a formula for the sequence de(cid:12)fined by a0 = 1, a1 = 2, and ak = 2a(subscript)k-1 + 3a(subscript)k-2, for all integers k is greater than or equal to 2. Solution characteristic equation is given by ak = 2ak-1+3ak-2 x^2-2x-3 = 0 solving the abovequadratic equation x = 2+sqrt(4-4*-3*1) / 2 or 2-sqrt(4-4*-3*1) / 2 x = 2+sqrt(16) / 2 or 2-sqrt(16) / 2 x = 2+4/2 or 2-4/2 x = 3 or -1 since roots are not equal resolved fromula given by ak = C.x1^k+ D.X2^k i.e ak = C.3^k+D.(-1)^k but given a0 = 1 and a1 = 2 thus a0 = 1 = C.3^0+D.(-1)^0 = C+D a1 = 2 = C.3^1+D.(-1)^1 = 3C-D C+D = 1 thus C = 1-D 3C-D = 2 solving we will get 3(1-D)-D = 2 --> 3-3D-D = 2 3-4D = 2 4D = 1 --> D= 1/4 C = 1-D = 1-1/4 = 3/4 thus formula for sequence given by ak = (3/4).3^k +(1/4)(-1)^k ak = (3k+1 + (-1)k) / 4 thus ans..

1. Find a formula for the sequence defined by a0 = 1, a1 = 2, and ak = 2a(subscript)k-1 +
3a(subscript)k-2, for all integers k is greater than or equal to 2.
Solution
characteristic equation is given by
ak = 2ak-1+3ak-2
x^2-2x-3 = 0
solving the abovequadratic equation
x = 2+sqrt(4-4*-3*1) / 2 or 2-sqrt(4-4*-3*1) / 2
x = 2+sqrt(16) / 2 or 2-sqrt(16) / 2
x = 2+4/2 or 2-4/2
x = 3 or -1
since roots are not equal resolved fromula given by
ak = C.x1^k+ D.X2^k
i.e ak = C.3^k+D.(-1)^k
but given a0 = 1 and a1 = 2
thus
a0 = 1 = C.3^0+D.(-1)^0 = C+D
a1 = 2 = C.3^1+D.(-1)^1 = 3C-D
C+D = 1 thus C = 1-D
3C-D = 2
solving we will get 3(1-D)-D = 2 --> 3-3D-D = 2
3-4D = 2
4D = 1 --> D= 1/4
C = 1-D = 1-1/4 = 3/4
thus
formula for sequence given by ak = (3/4).3^k +(1/4)(-1)^k
ak = (3k+1 + (-1)k) / 4
thus ans.