Limits, Continuity & Differentiability
General
Grade 12
Question:
<p>Let a = min{x2 + 2x + 3, x ∈R} and b = limθ→0 1−cos θ
θ2
. Then the value of Pn
r=0 arbn−r is:</p>
2n+1 -1\n1
2n+1 + 1\n1
4n+1 -1\n3 \cdot 2n
4n-1 -1\n3
Step-by-Step Solution
Key Concept: General
<p><strong>1</strong>: f(x) = x2 + 2x + 3 = (x + 1)2 + 2. The minimum value is a = 2 at x = -1.</p><p><strong>2</strong>: b = lim\theta\to 0 1-cos \theta</p> \theta2 = 1/2.<p><strong>3</strong>: Sum S = Pn</p> r=0 2r(1/2)n-r = (1/2)n Pn r=0(2/0.5)r = (1/2)n Pn r=0 4r.<p><strong>4</strong>: Using GP sum formula: S =</p> 1 2n 4n+1-1 4-1 = 4n+1-1 3 \cdot 2n .
Correct Answer: 3