If $a, b, c, d, e$ are five consecutive odd numbers, their average is

### Concept & Logic The average of any sequence of consecutive numbers in an Arithmetic Progression (AP) with an odd number of terms is always the exactly middle term. ### Step-by-Step Solution * **Given:** * $a, b, c, d, e$ are five consecutive odd numbers. * Since they are consecutive odd numbers, they have a common difference of $2$. * **Deduction:** * We can express all terms in relation to the first term $a$: * $b = a + 2$ * $c = a + 4$ * $d = a + 6$ * $e = a + 8$ * **Calculation:** * The average is the middle term, which is $c$. * Substituting the value of $c$, the average is $a + 4$. * Looking at the given options: * Option (a) is $5(a + 4)$, which represents the sum, not the average. * Options (b) and (c) are incorrect formulas. * Since $a + 4$ is not listed, the answer is "None of these". ### Exam Strategy & Shortcut For consecutive sequences (even, odd, or regular integers), the average is simply the median. The median of $a, b, c, d, e$ is immediately $c$. Since $c$ is two steps away from $a$ in an odd sequence, it is $a + 2(2) = a + 4$. ### Common Pitfall Mistaking the sum for the average. Many students will quickly calculate the average as $a+4$, see $5(a+4)$ in option (a), and select it in a rush, forgetting that multiplying by $5$ gives the total sum of the terms. ### Final Answer **Therefore, the correct answer is None of these.**