The average of five consecutive natural numbers is 50. What is the product of the smallest and the largest of the numbers?

Question

Accepted Answer

Let the first number of the five consecutive natural numbers be $$x$$. The average of these numbers is given as $$50$$, and the sum of the five numbers is:  
$$
50 	imes 5 = 250.
$$  
The five consecutive numbers are $$x, x+1, x+2, x+3, x+4$$. Their sum is:  
$$
x + (x+1) + (x+2) + (x+3) + (x+4) = 5x + 10.
$$  
Equating this to $$250$$:  
$$
5x + 10 = 250 \implies 5x = 240 \implies x = 48.
$$

Thus, the five consecutive numbers are $$48, 49, 50, 51, 52$$. The smallest number is $$48$$, and the largest number is $$52$$. Their product is:  
$$
48 	imes 52 = (50 - 2)(50 + 2) = 50^2 - 2^2 = 2500 - 4 = 2496.
$$

Therefore, the product of the smallest and largest numbers is **2496**.

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