If the sum of five consecutive even numbers is equal to the product of

If the sum of five consecutive even numbers is equal to the product of first five natural numbers, then which one of the following is the largest of those even numbers ?

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UPSC CISF-AC-EXE – 2022

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The product of the first five natural numbers (1, 2, 3, 4, 5) is $1 \times 2 \times 3 \times 4 \times 5 = 120$. Let the five consecutive even numbers be represented as $x-4, x-2, x, x+2, x+4$, where $x$ is the middle number. The sum of these five consecutive even numbers is $(x-4) + (x-2) + x + (x+2) + (x+4) = 5x$. We are given that this sum is equal to the product of the first five natural numbers, so $5x = 120$. Solving for $x$, we get $x = 120 / 5 = 24$. The five consecutive even numbers are $24-4=20$, $24-2=22$, $24$, $24+2=26$, and $24+4=28$. The largest of these even numbers is 28.

– First five natural numbers are 1, 2, 3, 4, 5. Their product is 120.
– Consecutive even numbers differ by 2. Representing them around the middle term simplifies the sum calculation.
– If the middle term is $x$, the sum of five consecutive even numbers is $5x$.

Using $x-4, x-2, x, x+2, x+4$ simplifies the sum calculation as the constant terms cancel out. Alternatively, if the first even number is $a$, the numbers are $a, a+2, a+4, a+6, a+8$. Their sum is $5a+20$. So $5a+20 = 120$, $5a = 100$, $a=20$. The numbers are 20, 22, 24, 26, 28. The largest is $a+8 = 20+8 = 28$.

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