A sequence is recursively defined by

Question

A sequence is recursively defined by 

k_n=k_n-1²+k_n-2, for n>2. k₁=1, k₂=0. What is the value of k₇?

the two is supposed to be on top of n-1 not squared. The answer is 27 but I don't know how to get there.

Answer 1

I think the answer can be obtained by simply following the recursion law for 5 steps:  Each term is the sum of the square of the previous one plus the value of the one two back.

 

Since  k1 = 1   and k2 = 0      k3 =   1² + 0  =  1

 

similarly   k4  =   1² + 0 = 1

 

and    k5   = 1² + 1  = 2

 

and     k6  =  2²  + 1 =   5

 

finally  k7 =   5²  + 2  =  27