The difference between the sum of the first 2n natural numbers and the sum of the first n
odd natural numbers is
Options:
(A) Which of the following statements can be inferred from the above passage?
commercial banks.
Decrease in repo rate will increase cost of borrowing and decrease lending by
(B) Increase in repo rate will decrease cost of borrowing and increase lending by
commercial banks.
(C) Increase in repo rate will decrease cost of borrowing and decrease lending by
(D) commercial banks.
Decrease in repo rate will decrease cost of borrowing and increase lending by
commercial banks.
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2020 Q.5)**
**1. Governing Concept**:
- **Topic**: Quantitative Aptitude & Arithmetic Progressions
- Sum of the first $2n$ natural numbers:
$S_1 = \sum_{k=1}^{2n} k = \frac{(2n)(2n + 1)}{2} = n(2n + 1) = 2n^2 + n$
- Sum of the first $n$ odd natural numbers ($1, 3, 5, \dots, 2n - 1$):
$S_2 = \sum_{k=1}^n (2k - 1) = n^2$
- Difference between the two sums:
$\Delta = S_1 - S_2 = (2n^2 + n) - n^2 = n^2 + n$
**2. Option Analysis**:
- **Option (A) [n^2 - n] is INCORRECT**: Sign error on linear term.
- **Option (B) [n^2 + n] is CORRECT**: Directly matches $(2n^2 + n) - n^2 = n^2 + n$.
- **Option (C) [2n^2 - n] is INCORRECT**: Fails to subtract $n^2$.
- **Option (D) [2n^2 + n] is INCORRECT**: Equals $S_1$ without subtracting the sum of odd numbers.
**3. Final Answer**:
- **Correct Choice**: **B**
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