$A, B, C$ and $D$ are four consecutive even numbers respectively and their average is 65. What is the product of $A$ and $D$?

### Concept & Logic When a sequence has an *even* number of terms, the median (and thus the average) falls exactly halfway between the two middle terms. ### Step-by-Step Solution * **Given:** * $A, B, C, D$ are four consecutive even numbers. * Their average is $65$. * **Deduction:** * The middle two terms are $B$ and $C$. * The average of the sequence ($65$) must be exactly halfway between $B$ and $C$. * Since they are consecutive even numbers, $B$ and $C$ must be the even numbers immediately before and after $65$. * Therefore, $B = 64$ and $C = 66$. * **Calculation:** * Now find $A$ and $D$. * $A$ is the even number before $B$: $A = 62$. * $D$ is the even number after $C$: $D = 68$. * The question asks for the product of $A$ and $D$. * $\text{Product} = 62 \times 68$ * $\text{Product} = 62 \times (70 - 2) = 4340 - 124 = 4216$. ### Exam Strategy & Shortcut Use the difference of squares shortcut. You know the average is $65$. $A$ and $D$ are symmetrically distributed around the average. $A$ is $3$ steps below ($65 - 3$) and $D$ is $3$ steps above ($65 + 3$). Product = $(65 - 3)(65 + 3) = 65^2 - 3^2$. $65^2 = 4225$. $4225 - 9 = 4216$. This is incredibly fast if you know your squares ending in 5. ### Common Pitfall Assuming that because the numbers are even, the average must also be an even number. If you get confused by an odd average ($65$) for even numbers, remember that the sum of 4 even numbers is even, but dividing an even sum by 4 can absolutely yield an odd number (e.g., $260 / 4 = 65$). ### Final Answer **Therefore, the correct answer is 4216.**