Questions: Rewrite this equation using telescoping methods: H=4-(7/(4-(7/(4-(7/(4-(7/...))))))) Read Example 3. H=7+(H/4) H=4+(7/H) H=7-(4/H) H=4-(7/H)

Solution Steps

To rewrite the given equation using telescoping methods, we need to recognize the pattern in the continued fraction and express it in a simpler form. We can set up an equation where \( H \) is equal to the continued fraction and solve for \( H \).

We start by setting up the equation based on the given continued fraction: \[ H = 4 - \frac{7}{4 - \frac{7}{4 - \frac{7}{4 - \ldots}}} \] We can express this as: \[ H = 4 - \frac{7}{4 - \frac{7}{H}} \]

To solve for \( H \), we rearrange the equation: \[ H = 4 - \frac{7}{4 - \frac{7}{H}} \] Multiplying both sides by \( H \) to eliminate the fraction, we get: \[ H \left(4 - \frac{7}{H}\right) = 4H - 7 \] Simplifying, we obtain: \[ H^2 = 4H - 7 \] Rearranging terms, we get a quadratic equation: \[ H^2 - 4H + 7 = 0 \]

We solve the quadratic equation \( H^2 - 4H + 7 = 0 \) using the quadratic formula: \[ H = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = -4 \), and \( c = 7 \). Plugging in these values, we get: \[ H = \frac{4 \pm \sqrt{16 - 28}}{2} \] \[ H = \frac{4 \pm \sqrt{-12}}{2} \] \[ H = \frac{4 \pm 2i\sqrt{3}}{2} \] \[ H = 2 \pm i\sqrt{3} \]

Final Answer