Biomolecules, enzyme kinetics

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2023 Q.32)** **1. Governing Concept & Definition**: - **Topic**: General Aptitude & Sum of Arithmetic Progression - The given series $S = 1 + 3 + 5 + 7 + 9 + \dots$ is the sequence of consecutive odd natural numbers. - For an AP with first term $a = 1$ and common difference $d = 2$, the sum of the first $n$ odd positive integers is: $S_n = n^2$ **2. Detailed Step-by-Step Calculation**: - **Given**: - Number of terms $n = 50$ - First term $a = 1$ - Common difference $d = 2$ - **Standard AP Sum Formula**: $S_n = \frac{n}{2} [2a + (n - 1)d] = \frac{50}{2} [2(1) + (50 - 1)(2)] = 25 \times [2 + 98] = 25 \times 100 = 2500$ - Or directly: $S_{50} = 50^2 = 2500$ **3. Final Answer**: - **Calculated Value**: **2500** - **Official GATE Accepted Range**: **2500 to 2500**