Question

A) Using Ramanjan's method, obtain the first six convergents of the equation. x+x² = 1

Answer 1

Ramanujan's method can be used to find the first six convergents of the quadratic equation x+x² = 1 using its continued fraction representation.

Step-by-step explanation:

Ramanujan's method can be used to find the convergents of a quadratic equation of the form x+x² = 1. We start by writing the equation as x² + x - 1 = 0. The quadratic equation can be solved using the quadratic formula: x = (-1 ± √5) / 2. Since the quadratic equation has two roots, we can find the continued fraction representation of each root and obtain the convergents. For example, for the positive root, x = (-1 + √5) / 2, the first six convergents can be obtained as follows:

1. Convergent 1: -1/2
2. Convergent 2: -1/2 + 1/(2+1/2)
3. Convergent 3: -1/2 + 1/(2+1/(2+1/2))
4. Convergent 4: -1/2 + 1/(2+1/(2+1/(2+1/2)))
5. Convergent 5: -1/2 + 1/(2+1/(2+1/(2+1/(2+1/2)))))
6. Convergent 6: -1/2 + 1/(2+1/(2+1/(2+1/(2+1/(2+1/2)))))

Learn more about Ramanujan's method for finding convergents of quadratic equations