Question: Within the set of whole numbers, one set of numbers satisfies the recursive formula \( a_n =…

Within the set of whole numbers, one set of numbers satisfies the recursive formula \( a_n = (a_{n-1})^2 - a_{n-1} \), when \( a_4 = 870 \). What are the first two terms of the sequence?

6, 30 2, 3 3, 6 2, 6

Solution

To solve this problem, we need to find the first two terms of the sequence based on the recursive formula: \[ a_n = (a_{n-1})^2 - a_{n-1} \] We know that \(a_4 = 870\). Let’s find the sequence backward starting from \(a_4\) and determine the first two terms. Start with \(n = 4\): 1. Given \(a_4 = 870\), use: \[ a_4 = (a_3)^2 - a_3 = 870 \] Rewrite the equation as: \[ (a_3)^2 - a_3 - 870 = 0 \] This is a quadratic equation: \[ a_3^2 - a_3 - 870 = 0 \] 2. Use the quadratic formula, \(a = 1\), \(b = -1\), \(c = -870\): \[ a_3 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{1 \pm \sqrt{1 + 4 \times 870}}{2} \] Simplify: \[ a_3 = \frac{1 \pm \sqrt{3481}}{2} \] Since \(\sqrt{3481} = 59\): \[ a_3 = \frac{1 \pm 59}{2} \] Calculate the two possible values for \(a_3\): \[ a_3 = \frac{60}{2} = 30 \quad \text{or} \quad a_3 = \frac{-58}{2} = -29 \] Since \(a_3\) must be a whole number, \(a_3 = 30\). 3. Use \(a_3 = 30\) to find \(a_2\): \[ a_3 = (a_2)^2 - a_2 = 30 \] Rewrite as a quadratic equation: \[ a_2^2 - a_2 - 30 = 0 \] 4. Use the quadratic formula again: \[ a_2 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{1 \pm \sqrt{1 + 4 \times 30}}{2} \] Simplify: \[ a_2 = \frac{1 \pm \sqrt{121}}{2} \] Since \(\sqrt{121} = 11\): \[ a_2 = \frac{1 \pm 11}{2} \] Calculate the two possible values for \(a_2\): \[ a_2 = \frac{12}{2} = 6 \quad \text{or} \quad a_2 = \frac{-10}{2} = -5 \] Since \(a_2\) must be a whole number, \(a_2 = 6\). 5. Finally, use \(a_2 = 6\) to find \(a_1\): \[ a_2 = (a_1)^2 - a_1 = 6 \] Rewrite as a quadratic equation: \[ a_1^2 - a_1 - 6 = 0 \] 6. Use the quadratic formula: \[ a_1 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{1 \pm \sqrt{1 + 4 \times 6}}{2} \] Simplify: \[ a_1 = \frac{1 \pm \sqrt{25}}{2} \] Since \(\sqrt{25} = 5\): \[ a_1 = \frac{1 \pm 5}{2} \] Calculate the two possible values for \(a_1\): \[ a_1 = \frac{6}{2} = 3 \quad \text{or} \quad a_1 = \frac{-4}{2} = -2 \] Since \(a_1\) must be a whole number, \(a_1 = 3\). Thus, the first two terms of the sequence are \(3, 6\).