Math Answers

A father and his son entered a competition, the probability that the father wins a prize is 1/7, and that his son wins a prize is 2/5. Find the probability that

(1). Either of them (but not both) wins a prize.

(2). At least one of them wins a prize.

Let 𝑃𝑃(𝑛𝑛) be the proposition that 1(1!) + 2(2!) + 3(3!) + ⋯+ 𝑛𝑛(𝑛𝑛!) = (𝑛𝑛+ 1)! −1. Prove by induction that 𝑃𝑃(𝑛𝑛) is true for all 𝑛𝑛≥1.

Prove that the product of any three consecutive integers is a multiple of 3. 

Use a proof by contraposition to show that if 𝑛𝑛2 + 1 is even, then 𝑛𝑛 is odd.

Use a direct proof to show that every odd integer is the difference of two squares.  

A man entered an orchard through 7 gates, and there took a certain number of apples.

When he left the orchard, he gave the first guard half of the apples that he had and 1

apple more. To the guard at the second gate, he gave half of his remaining apples and 1

apple more. He did the same with each of the remaining 5 guards and left the orchard

with 1 apple. How many apples did he gather in the orchard?

2. Use generating functions to solve the recurrence relation a_n = 4a_n−1 − 4a_n−2 + n², where a₀ = 2, a₁ = 5.

Show that the transformation x = z −"\\frac{b}{3a}" converts the cubic equation ax³ + bx² + cx + d= 0 into one of the form z³ + 3Hz + G = 0, i.e. the x² term goes away.

If F_n is the n-th Fibonacci number, show that lim "\\lim_{n \\to \\infty} \\frac { F_{n+1}}{F_n} = \\frac{1+\\sqrt{5}}{2}"

In a family of 11 children, what is the probability that there will be more boys than girls?