Given the sequence $a_{0} = 1, a_{1} = 2, a_{k + 1} = a_{k} + \frac{a_{k - 1}}{1 + a_{k - 1}^{2}}, k > 1$ $a_0=1,a_1=2, a_(k+1)=a_k+(a_(k-1))/(1+a_(k-1)^2), k > 1$ for what value of $k$ $k$ occours $52 < a_{k} < 65$ $52 < a_k < 65$ ?

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Given the sequence $a_{0} = 1, a_{1} = 2, a_{k + 1} = a_{k} + \frac{a_{k - 1}}{1 + a_{k - 1}^{2}}, k > 1$ $a_0=1,a_1=2, a_(k+1)=a_k+(a_(k-1))/(1+a_(k-1)^2), k > 1$ for what value of $k$ $k$ occours $52 < a_{k} < 65$ $52 < a_k < 65$ ?

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Answer 1 · 2017-10-12T21:43:50

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 $a_{k}$ $a_k$ . 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 $52 < a_{k} < 65$ $52 < a_k < 65$ is:

$k = 1350, a_{k} = 52.01277$ $k = 1350, a_k = 52.01277$

The last value of k where $52 < a_{k} < 65$ $52 < a_k < 65$ is:

$k = 2109, a_{k} = 64.98885$ $k = 2109, a_k = 64.98885$

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Given the sequence $a_{0} = 1, a_{1} = 2, a_{k + 1} = a_{k} + \frac{a_{k - 1}}{1 + a_{k - 1}^{2}}, k > 1$ $a_0=1,a_1=2, a_(k+1)=a_k+(a_(k-1))/(1+a_(k-1)^2), k > 1$ for what value of $k$ $k$ occours $52 < a_{k} < 65$ $52 < a_k < 65$ ?

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Given the sequence a0=1,a1=2,ak+1=ak+ak−11+a2k−1,k>1a_0=1,a_1=2, a_(k+1)=a_k+(a_(k-1))/(1+a_(k-1)^2), k > 1 for what value of kk occours 52<ak<6552 < a_k < 65 ?

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Given the sequence $a_{0} = 1, a_{1} = 2, a_{k + 1} = a_{k} + \frac{a_{k - 1}}{1 + a_{k - 1}^{2}}, k > 1$ $a_0=1,a_1=2, a_(k+1)=a_k+(a_(k-1))/(1+a_(k-1)^2), k > 1$ for what value of $k$ $k$ occours $52 < a_{k} < 65$ $52 < a_k < 65$ ?