Question

The average of 11 consecutive odd numbers is 15. What is the sum of the second and the tenth number?

• A. 12
• B. 24
• C. 26
• D. 30

Hint:

For problems involving consecutive numbers, use the properties of arithmetic sequences to simplify calculations.

Answer 1

The Correct Option is D

Let the 11 consecutive odd numbers be represented as: \[ a, a+2, a+4, \dots, a+20 \] The average of these numbers is 15, so we have: \[ \frac{a + (a+2) + (a+4) + \dots + (a+20)}{11} = 15 \] The sum of the terms in the series \( a, a+2, a+4, \dots, a+20 \) is an arithmetic progression. The sum of 11 terms of an arithmetic progression is given by: \[ \frac{n}{2} \times (\text{first term} + \text{last term}) = \frac{11}{2} \times (a + (a+20)) = \frac{11}{2} \times (2a + 20) \] This gives the equation: \[ \frac{11}{2} \times (2a + 20) = 15 \times 11 = 165 \] Simplifying: \[ 11 \times (a + 10) = 165 \quad \Rightarrow \quad a + 10 = 15 \quad \Rightarrow \quad a = 5 \] Thus, the numbers are: \[ 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25 \] The second number is \(7\), and the tenth number is \(23\). The sum of the second and tenth numbers is: \[ 7 + 23 = 30 \]