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)
• Thirteen consecutive odd numbers form an arithmetic sequence with common difference 2: a, a+2, a+4, …, a+24. The first five terms are a, a+2, a+4, a+6, a+8, and their mean is (5a + 20)/5 = a + 4.
• Setting a + 4 = 39 gives a = 35, so the thirteen numbers are 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59.
• For any arithmetic sequence, the mean equals the middle term. With thirteen terms, the middle term is the 7th term.
• The 7th term is a + 2×6 = 35 + 12 = 47.
• Verification: the sum of all thirteen numbers is 13 × (35 + 59)/2 = 13 × 47 = 611, and 611 ÷ 13 = 47.
• Therefore the mean of all thirteen numbers is 47, which is option (a).
Why the other options are wrong
- (b) 49
- 49 is the 8th term in the sequence, not the 7th (middle) term; it results from miscounting the middle position of thirteen terms as the 8th rather than the 7th.
- (c) 51
- 51 is the 9th term (35 + 2 × 8), not the 7th (middle) term, which is 47.
- (d) 45
- 45 is the 6th term in the sequence, not the 7th (middle) term; it results from using a + 10 instead of a + 12 to locate the middle of thirteen terms.
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.