The sum of 6 consecutive integers is 393 . What is the third number in this sequence?

To find the third number in a sequence of 6 consecutive integers that sum to 393, we start by letting the first integer be \( x \). Thus, the integers can be expressed as \( x, x+1, x+2, x+3, x+4, x+5 \). Their sum is: \[ x + (x+1) + (x+2) + (x+3) + (x+4) + (x+5) = 6x + 15 \] Setting this equal to 393 gives: \[ 6x + 15 = 393 \] Subtracting 15 from both sides results in: \[ 6x = 378 \] Dividing both sides by 6, we find: \[ x = 63 \] Thus, the 6 consecutive integers are: 63, 64, 65, 66, 67, 68. The third number in this sequence is \( \textbf{65} \). Now, let’s explore a fun fact: these integers are evenly distributed, making them an excellent example of simple arithmetic properties. And if you were to visualize it as a classic number line, they would neatly form a cluster right around the value of 65, showcasing how symmetry works in mathematics. Also, engaging with sequences and series has many real-world applications, such as predicting trends in data or modeling patterns in nature. Understanding these sums can help in various fields including economics, programming, and even game development, where sequences might represent scores or resources.