--- title: "Fibonacci difference equation: series to 8 terms" subject: "Digital Control System" chapter: 2 key: "semester_6/digital_control_system:fibonacci-series-8-terms" source: https://www.bctengineer.com/q/semester_6/digital_control_system/fibonacci-series-8-terms site: BCT Engineering --- # Fibonacci difference equation: series to 8 terms Digital Control System · Chapter 2: The Z-Transform - Asked in: 2079 Chaitra, 8 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/digital_control_system/fibonacci-series-8-terms - Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-2 ## Question Solve the difference equation x(k+2)-x(k+1)-x(k)=0, where x(0)=0 and x(1)=1, and obtain the general form of solution. After obtaining the general form of solution determine the series up to 8th term. ## Short answer Take z-transform: X(z)=z/(z^2-z-1). Partial fractions give x(k)=(1/√5)[((1+√5)/2)^k - ((1-√5)/2)^k]. The first eight terms (k=0 to 7) are 0, 1, 1, 2, 3, 5, 8, 13. ## Answer This is the **Fibonacci** difference equation. Solve it by z-transform, then list the terms. #### Step 1: Take the z-transform Use $\mathcal{Z}[x(k+2)] = z^2X(z) - z^2x(0) - zx(1)$ and $\mathcal{Z}[x(k+1)] = zX(z) - zx(0)$, with $x(0) = 0$ and $x(1) = 1$: $$ \left[z^2X(z) - z\right] - zX(z) - X(z) = 0 $$ $$ X(z) = \frac{z}{z^2 - z - 1} $$ #### Step 2: Factorise the denominator $$ z^2 - z - 1 = 0 \;\Rightarrow\; z_{1,2} = \frac{1 \pm \sqrt{5}}{2}, \quad z_1 = 1.6180,\; z_2 = -0.6180 $$ #### Step 3: Partial fractions of $X(z)/z$ $$ \frac{X(z)}{z} = \frac{1}{(z-z_1)(z-z_2)} = \frac{1}{z_1 - z_2}\left[\frac{1}{z-z_1} - \frac{1}{z-z_2}\right], \quad z_1 - z_2 = \sqrt{5} $$ $$ X(z) = \frac{1}{\sqrt5}\left[\frac{z}{z - z_1} - \frac{z}{z - z_2}\right] $$ #### Step 4: General (closed-form) solution $$ x(k) = \frac{1}{\sqrt5}\left[\left(\frac{1+\sqrt5}{2}\right)^k - \left(\frac{1-\sqrt5}{2}\right)^k\right], \quad k = 0, 1, 2, \dots $$ Here $1/\sqrt5 = 0.4472$. Check: $x(0) = 0$, and $x(1) = \frac{1}{\sqrt5}\cdot\sqrt5 = 1$. #### Step 5: Series up to the 8th term From the closed form, or from the recursion $x(k+2) = x(k+1) + x(k)$: | Term | 1st | 2nd | 3rd | 4th | 5th | 6th | 7th | 8th | |---|---|---|---|---|---|---|---|---| | $k$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | | $x(k)$ | 0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | Example check with the formula for $k = 7$: $\frac{1}{\sqrt5}\left[(1.618)^7 - (-0.618)^7\right] = \frac{1}{2.2361}\left[29.034 + 0.034\right] = 13.0$. (If the series is counted from $k = 1$, the 8th term is $x(8) = 21$.) #### Remarks - One pole ($z = 1.618$) is outside the unit circle, so the sequence grows without bound. - For large $k$ the second term dies out, so $x(k) \approx 0.4472(1.618)^k$. The ratio of successive terms tends to the golden ratio $1.6180$. **Answer:** $x(k) = \dfrac{1}{\sqrt5}\left[\left(\dfrac{1+\sqrt5}{2}\right)^k - \left(\dfrac{1-\sqrt5}{2}\right)^k\right]$. Series: $0, 1, 1, 2, 3, 5, 8, 13$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. With x(0)=0, x(1)=1, the z-transform gives X(z) = ? A. z^2/(z^2 - z - 1) B. z/(z^2 + z - 1) C. z/(z^2 - z - 1) D. 1/(z^2 - z - 1) Answer: C. z^2X - z - zX - X = 0. (Key point: Transform of difference equation) 2. What are the roots of z^2 - z - 1 = 0? A. 1.618 and 0.618 B. -1.618 and 0.618 C. 2 and -1 D. 1.618 and -0.618 Answer: D. z = (1 +- sqrt5)/2. (Key point: Roots) 3. Which is the closed form? A. [(1+sqrt5)/2]^k - [(1-sqrt5)/2]^k all over sqrt5 B. The same with a plus sign C. The same without 1/sqrt5 D. [(1+sqrt5)/2]^k only Answer: A. Partial fractions give 1/sqrt5 [z/(z-z1) - z/(z-z2)]. (Key point: Closed form) 4. What are the first eight terms (k = 0 to 7)? A. 0, 1, 1, 2, 4, 7, 13, 24 B. 0, 1, 1, 2, 3, 5, 8, 13 C. 1, 1, 2, 3, 5, 8, 13, 21 D. 0, 1, 2, 3, 5, 8, 13, 21 Answer: B. Each term is the sum of the previous two. 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