GATE BT 2018 β€’ Q53 NAT β€’ 2 Marks

Section 1: Engineering Mathematics

Calculus

Subtopic: Fundamentals of Calculus

Question Statement:

If 1 + π‘Ÿ + π‘Ÿ2 + π‘Ÿ3 + β‹―βˆž = 1.5, then, 1 + 2π‘Ÿ + 3π‘Ÿ2 + 4π‘Ÿ3 + β‹―βˆž= (up to two decimal places) _________ GATE 2018 Biotechnology

Official Answer Key:

2.25Β toΒ 2.25

Step-by-Step Solution & Explanation

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2018 Q.53)** **1. Governing Concept & Definition**: - **Topic**: Engineering Mathematics & Infinite Geometric Series - Given $S_1 = 1 + r + r^2 + r^3 + \dots = \frac{1}{1 - r}$. - The derivative series is $S_2 = 1 + 2r + 3r^2 + 4r^3 + \dots = \frac{d}{dr}(r + r^2 + r^3 + \dots) = \frac{1}{(1 - r)^2}$. **2. Detailed Step-by-Step Calculation**: - **Given**: $\frac{1}{1 - r} = 1.5 = \frac{3}{2}$ $1 - r = \frac{2}{3} \implies r = \frac{1}{3}$ - **Evaluate Target Series $S_2$**: $S_2 = \frac{1}{(1 - r)^2} = \left(\frac{1}{1 - r}\right)^2 = (1.5)^2 = 2.25$ **3. Final Answer**: - **Calculated Value**: **2.25** - **Official GATE Accepted Range**: **2.25 to 2.25**

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