Q.5 The difference between the sum of the first 2n
natural numbers and the sum of the first
odd natural numbers is
(A) n2-n
(B) n2+n
(C) 2n2-n
(D) 2n2+n
The correct answer is (B) n2+n.
This multiple-choice question tests knowledge of standard summation formulas in arithmetic series.
This multiple-choice question tests knowledge of standard summation formulas in arithmetic series.
Key Formulas
Sum of first k natural numbers: k(k+1)/2
For first 2n naturals: 2n(2n+1)/2 = n(2n+1)
Sum of first n odd naturals (1 + 3 + … + (2n-1)): n2
For first 2n naturals: 2n(2n+1)/2 = n(2n+1)
Sum of first n odd naturals (1 + 3 + … + (2n-1)): n2
Correct Answer Explanation
Difference: n(2n+1) – n2 = 2n2 + n – n2 = n2 + n
Verification for n=1: Sum to 2 is 3, first odd sum is 1, difference 2 = 12 + 1
For n=2: Sum to 4 is 10, first 2 odds sum to 4, difference 6 = 4 + 2
Options Analysis
| Option | Expression | Matches? | Reason |
|---|---|---|---|
| (A) | n2-n | No | Equals n(n-1); for n=2, gives 2, but actual is 6. |
| (B) | n2+n | Yes | Simplifies exactly to derived difference. |
| (C) | 2n2-n | No | Larger; for n=2, gives 6, but mismatches pattern (e.g., n=1:1). |
| (D) | 2n2+n | No | Sum of first 2n naturals itself, ignores subtraction. |
