Question

Given the sequence `a_0=1,a_1=2, a_(k+1)=a_k+(a_(k-1))/(1+a_(k-1)^2), k > 1` for what value of `k` occours `52 < a_k < 65` ?

Answer 1

I used an Excel spreadsheet.

I used column A for my value of k. I entered 0 in cell A1 and I entered =A1+1 in cell A2; this will give me a column of integers incremented by 1 as I copy and block paste the column.

I used column B for my value of `a_k`. I entered 1 in cell B1, 2 in cell B2, and =B2+B1/(1+B1^2) in cell B3; this will give me the recursive calculation as I copy and block paste the column.

The first value of k where `52 < a_k < 65` is:

`k = 1350, a_k = 52.01277`

The last value of k where `52 < a_k < 65` is:

`k = 2109, a_k = 64.98885`