For the product of k consecutive integers, which of the following is true?

For the product of k consecutive integers, which of the following is true?

• A. It is always greater than k
• B. It may be less than k
• C. It is generally not related to k
• D. It can be equal to k

Answer: (A)

The product of \( k \) consecutive integers, denoted as \( n \times (n+1) \times (n+2) \times \ldots \times (n+k-1) \), will always yield a value greater than \( k \) when \( n \) begins at 1 or any positive integer. This is because the smallest product occurs starting from the first positive integer (1), where the product of, for example, 3 consecutive integers is \( 1 \times 2 \times 3 = 6 \), which is greater than 3. As \( n \) increases, the product increases even more significantly, especially as \( k \) grows larger than 2. The nature of consecutive integers, encompassing both minimum and maximum values within and extending beyond \( n \), generally ensures that the product accelerates beyond just \( k \). In cases where \( n \) starts from zero or is negative, the product may still exceed \( k \) in absolute value. For instance, if \( n = -1 \) and \( k = 2 \), the integers would be \( -1 \) and \( 0 \), resulting in a product of \(

The product of ( k ) consecutive integers, denoted as ( n \times (n+1) \times (n+2) \times \ldots \times (n+k-1) ), will always yield a value greater than ( k ) when ( n ) begins at 1 or any positive integer. This is because the smallest product occurs starting from the first positive integer (1), where the product of, for example, 3 consecutive integers is ( 1 \times 2 \times 3 = 6 ), which is greater than 3.

As ( n ) increases, the product increases even more significantly, especially as ( k ) grows larger than 2. The nature of consecutive integers, encompassing both minimum and maximum values within and extending beyond ( n ), generally ensures that the product accelerates beyond just ( k ).

In cases where ( n ) starts from zero or is negative, the product may still exceed ( k ) in absolute value. For instance, if ( n = -1 ) and ( k = 2 ), the integers would be ( -1 ) and ( 0 ), resulting in a product of (