There are thirteen 2-digit consecutive odd numbers. If 39 is the mean of the first five such numbers, then what is the mean of all the thirteen numbers?
- (a) 47 ✓ UPSC's answer
- (b) 49
- (c) 51
- (d) 45
Why the answer is (a)
• Let the first five consecutive odd numbers be x, x+2, x+4, x+6 and x+8.
• Their mean is the middle number, x+4, so x+4 = 39.
• Therefore x = 35, and the thirteen numbers are 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57 and 59.
• The mean of an arithmetic sequence is the middle term, the 7th number.
• The 7th number is 35 + 2 × 6 = 47, so the mean of all thirteen numbers is 47, option (a).
Why the other options are wrong
- (b) 49
- 49 is the mean of the last five numbers, 45, 47, 49, 51 and 53, not of all thirteen numbers.
- (c) 51
- 51 is the mean of the last three numbers, 49, 51 and 53, not of all thirteen numbers.
- (d) 45
- 45 is the sixth number in the sequence, not the seventh and middle number that gives the mean.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2017, held on 18 June 2017. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.