Binomial Theorem
Binomial Theorem
star_batch_jee_advanced_2025
Grade 11
Question:
Let $n$ be a positive integer and $(1 + x + x^2)^n = a_0 + a_1 x + a_2 x^2 + \ldots + a_{2n-1}x^{2n-1} + a_{2n}x^{2n}$, then:
1. \sum_{r=0}^{n-1} a_r = \frac{1}{2}(3^n - a_n)
2. \sum_{r=0}^{2n} (-1)^r a_r^2 = a_n
3. a_0^2 - a_1^2 + a_2^2 - a_3^2 + \ldots + (-1)^r a_{2n-1}^2 = \frac{a_n}{2}(1 - (-1)^n a_n)
4. (r+1)a_{r+1} = (n-r)a_r + (2n-r+1)a_{r-1}, 1 \leq r \leq 2n-1
Step-by-Step Solution
Key Concept: The minimum sum of distances from a point to two fixed points occurs when the point lies on the line segment connecting them, equaling the distance between those two points.
The sum of distances from $z$ to the origin and to point $(\cos\alpha, \sin\alpha)$ is minimized when $z$ lies on the line segment joining these two points. Since the origin is at distance $0$ and the point is at distance $1$ from origin, the minimum sum equals $\sqrt{(\cos\alpha - 0)^2 + (\sin\alpha - 0)^2} = 1$.
Correct Answer: I need to find which statement(s) about the expansion $(1 + x + x^2)^n = \sum_{r=0}^{2n} a_r x^r$ is/are correct.
Let me verify each option:
**Option 1:** $\sum_{r=0}^{n-1} a_r =