Consecutive Integers Calculator

Consecutive integers are numbers that follow each other in order, from smallest to largest, without gaps (such as 4, 5, 6). Algebraically, a sequence of $n$ consecutive integers starting from a base integer $x$ can be represented as:

Sum = x + (x + 1) + (x + 2) + ... + (x + n - 1)

For consecutive even or odd integers, the numbers increase by 2 instead of 1, represented as $x$, $x + 2$, $x + 4$, and so on. By setting up a linear equation equal to your target result, the calculator solves for the starting integer $x$ and generates the full sequence instantly.

Let us find three consecutive odd integers that add up to a total sum of 57. First, we define our variables. Let our first odd integer be $x$. Because consecutive odd integers skip by 2, our second integer is $x + 2$ and our third integer is $x + 4$.

Next, we set up our equation by adding them together and setting them equal to the target sum:

x + (x + 2) + (x + 4) = 57

Combine the like terms to simplify the expression:

3x + 6 = 57

Subtract 6 from both sides of the equation:

3x = 51

Divide by 3 to isolate $x$:

x = 17

Now that we know our starting integer is 17, our complete sequence of three consecutive odd integers is 17, 19, and 21. Checking our math: 17 + 19 + 21 equals 57.

When tackling word problems involving consecutive numbers, pay close attention to whether the prompt asks for standard integers, even integers, or odd integers. Remember that even and odd sequences both increase by increments of 2, meaning your step size is always 2 rather than 1. Always verify your final sequence by adding the generated numbers back together to ensure they match your target sum.