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Mathematical induction/pascal triangle

Please help me with the attached pre-calculus questions.

1. Use mathematical induction to prove that 6 is a factor of n (n + 1)(n + 2).

2. Use mathematical induction to prove 1^2 + 3^2 + 5^2 + . . . + (2n - 1)^2 = n (4n^2 - 1) / 3.

3. Expand (2x - y)^6 using binomial coefficients and then evaluate each coefficient.
Example: the first term is 6 chose 0 (2x)^6 (-y)^0 = 64x^6

4. Interpret what the result in problem 3 tells us about construction the next row of the Pascal triangle:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1

5. Find the equation of the circle that has its center at the focus of the parabola X^2 - 4x + 6y - 20 = 0 and that is tangent to the parabola at its vertex. Also sketch both curves.

6. Find the equation of the Ellipse whose foci coincide with the foci of the hyperbola x^2 - 3y^2 - 2x - 12y + 1 = 0 and whose eccentricity is the reciprocal of that of the hyperbola.

Solution Summary

This solution includes step by step solutions of problems on Mathematical Induction, pascal triangle and finding equation of ellipse and circle.

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