Question

Give a recursive definition of the sequence {a_n}, n = 1, 2, 3,... if

a_n = n²

^{커-1, an-an-1+2n, for all n>1 어=1, an = an-1+2n-1, for all n21 an = an-1+2n-1. for all n21 gel, an=an-1+2n-1, for all n>1 -1, an- an-1+2n-1, for all n2o}

Answer 1

The sequence is a_n = n^2 equation (1) for n = 1, a_1 = 1^2 = 1 n = 2, a_2 = 2^2 = 4 n = 3, a_3 = 9 = 9 n = 4, a_4 = 4^2 = 16 For n^th term a_n = n^2 for (n + 1)^th term a_n + 1 = (n + 1)^2 = n^2 + 2n + 1 a_n + 1 = a_n + 2n + 1 [because From (1)] replace n = n - 1 a_n - 1 + 1 = a_n - 1 + 2(n - 1) + 1 a_n = a_n - 1 + 2n - 2 + 1 a_n = a_n - 1 + 2n - 1, a_1 = 1 therefore Recursive definition of the sequence is a_1 = 1, a_n = a_n - 1 + 2n - 1 n greaterthanorequalto 1 so option 'B' is the correct