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)
• The expression is $10^n + 1$. For the sum of digits to be 2, the number must consist of a '1' followed by zeros, with the last digit being '1' (e.g., 11, 101, 1001), or simply '2' if no zeros are present.
• If $n$ is a whole number (non-negative integer: 0, 1, 2, ...), $10^n$ is an integer power of 10, which is 1 followed by $n$ zeros.
• Adding 1 to $10^n$ results in a number with a leading 1, $n-1$ zeros in the middle (if $n > 1$), and a trailing 1. For example, $10^0 + 1 = 2$ (sum 2), $10^1 + 1 = 11$ (sum 2), $10^2 + 1 = 101$ (sum 2).
• In all these cases where $n$ is a whole number, the decimal representation contains only the digits 1 and 0, specifically two 1s and the rest 0s, making the sum of digits exactly 2.
• Therefore, the condition holds for any whole number $n$.
Why the other options are wrong
- (a) For n = 0 only
- The condition holds for $n=1$ (11) and $n=2$ (101) as well, not just $n=0$.
- (c) For any positive integer n only
- The condition also holds for $n=0$ (which is a whole number but not a positive integer), so the restriction to positive integers is incorrect.
- (d) For any real number n
- If $n$ is a non-integer real number (e.g., 0.5), $10^n$ is not an integer, and the sum of digits is not defined in the standard integer sense or does not equal 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.