JEE Advance - Mathematics (2000 - No. 12)
Prove by induction on, that $${p_n} = A{\alpha ^n} + B{\beta ^n}$$ for all $$n \ge 1,$$ where $$\alpha $$ and $$\beta $$ are the roots of quadratic equation $${x^2} - \left( {1 - p} \right)x - p\left( {1 - p} \right) = 0$$ and $$A = {{{p^2} + \beta - 1} \over {\alpha \beta - {\alpha ^2}}},B = {{{p^2} + \alpha - 1} \over {\alpha \beta - {\beta ^2}}}.$$
The problem asks to prove a formula for the probability of no consecutive heads in n coin tosses using induction, given a recurrence relation and initial values. The formula involves roots of a quadratic equation and constants A and B. The task is essentially to verify that the given formula satisfies the recurrence and initial conditions.
The problem requires calculating the probability of getting exactly one head in n coin tosses.
The problem requires calculating the probability of getting at least one head in n coin tosses.
The problem requires calculating the probability of getting exactly two heads in n coin tosses.
The problem requires calculating the probability of getting at most one head in n coin tosses.
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