Binomial Theorem
Fractional part; parity via binomial theorem
Grade Class 12
Question:
Let $\left[(5+\sqrt{b})^n\right] = N$ where $b$, $n$ are natural numbers and $|5-\sqrt{b}|<1$. If $b$ and $n$ are picked randomly, then the probability that $N$ is odd belongs to the set (where $[\cdot]$ denotes the greatest integer function)
$\left[\frac{3}{4}, 1\right]$
$\left\{\frac{3}{4}\right\}$
$\left[\frac{1}{2}, \frac{3}{4}\right]$
None of these
Step-by-Step Solution
Key Concept: Use the identity: $(5+\sqrt{b})^n + (5-\sqrt{b})^n$ is always an integer (binomial expansion). Since $0 < 5-\sqrt{b} < 1$, the fractional part and parity of $N=[(5+\sqrt{b})^n]$ depends on parity of $n$.
When $n$ even, $N$ is always odd. When $n$ odd, $N$ is odd only for specific $b$ values ($b=17$ to $24$ out of valid range). Probability $\in [1/2, 3/4]$.
Correct Answer: 3