Answer to Question #241818 in Math for Jai

Question

1. What is the 50th term of the fibonacci sequence?

2. Solve for the 67th term of the fibonacci sequence?

3. The 80th term of fibonacci sequence is

4. What is the 99th term of fibonacci sequence?

5. The 100th term of fibonacci sequence is

Answer 1

Use Binet formula

$$F_n=\dfrac{\alpha^n-\beta^n}{\alpha-\beta}$$

$$\alpha=\dfrac{1+\sqrt{5}}{2}, \beta=\dfrac{1-\sqrt{5}}{2}$$

$$\alpha-\beta=\sqrt{5}$$

1.

$$F_{50}=\dfrac{(1+\sqrt{5})^{50}-(1-\sqrt{5})^{50}}{2^{50}\sqrt{5}}$$

$$F_{50}= 12586269025$$

2.

$$F_{67}=\dfrac{(1+\sqrt{5})^{67}-(1-\sqrt{5})^{67}}{2^{67}\sqrt{5}}$$

$$F_{67}= 44945570212853$$

3.

$$F_{80}=\dfrac{(1+\sqrt{5})^{80}-(1-\sqrt{5})^{80}}{2^{80}\sqrt{5}}$$

$$F_{80}= 23416728348467685$$

4.

$$F_{99}=\dfrac{(1+\sqrt{5})^{99}-(1-\sqrt{5})^{99}}{2^{99}\sqrt{5}}$$

$$F_{99}= 218922995834555169026$$

5.

$$F_{100}=\dfrac{(1+\sqrt{5})^{100}-(1-\sqrt{5})^{100}}{2^{100}\sqrt{5}}$$

$$F_{100}= 354224848179261915075$$