Question

Solve for the 67th term of the fibonacci sequence?

Answer 1

Use Binet formula

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

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

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

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

$$F_{67}=27777890035288$$