For what value of n, the sum of digits in the number (10ⁿ + 1) is 2?
- (a) For n = 0 only
- (b) For any whole number n ✓ UPSC's answer
- (c) For any positive integer n only
- (d) For any real number n
Why the answer is (b)
• For n = 0: 10⁰ + 1 = 2, digit sum 2.
• For n = 1, 2, 3, …: 10ⁿ + 1 = 11, 101, 1001, …, always two 1s and zeros, so the digit sum is 2.
• For a negative or fractional n, 10ⁿ + 1 is not a whole number (e.g. 1·1 or about 4·16), so 'for any real number' is wrong.
• The condition holds for every whole number n (0, 1, 2, …) — option (b).
Why the other options are wrong
- (a) For n = 0 only
- The condition also holds for n = 1 (11), n = 2 (101) and so on, not just n = 0.
- (c) For any positive integer n only
- It also holds for n = 0 (10⁰ + 1 = 2), which is a whole number but not a positive integer.
- (d) For any real number n
- For a fractional n such as 0·5, 10ⁿ + 1 is not a whole number, so the digit sum is not 2.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2020, held on 4 October 2020. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.