Can consecutive integers be negative?

Yes. Consecutive integers can start at any integer—positive, negative, or zero. For example, -5, -4, -3 are three consecutive integers. The algebraic method is identical whether the integers are positive or negative.

What is the difference between consecutive integers and consecutive even or odd integers?

Consecutive integers (n, n+1, n+2, \ldots) include every whole number in a row. Consecutive even integers or consecutive odd integers (n, n+2, n+4, \ldots) skip every other number. Use the second pattern only when the problem explicitly says 'even' or 'odd.'

Do I need to substitute back, or is n the answer?

You must substitute back. The variable n represents only the first integer; the problem usually asks for all the integers in the sequence. If n = 7 and you need three consecutive integers, your answer is 7, 8, and 9—not just 7.

Can I use a different variable or starting point?

Yes. Some people use n-1, n, n+1 (with n in the middle) or other setups. However, using n as the first integer keeps the arithmetic simpler and requires fewer parentheses, making it the standard approach.

Consecutive Integer Problems

Set up equations where unknowns represent consecutive integers to find specific number sequences meeting given conditions.

n, ; n+1, ; n+2, \ldots

Solve a problem with Consecutive Integer Problems

Type the problem. The solver will use Consecutive Integer Problems where Consecutive Integer Problems is the right tool, and tell you when it is not.

What each symbol means

What Consecutive Integer Problems takes

Symbol	Meaning
$n$	The first integer in the sequence; treating $n$ as a non-integer or misplacing it (for example, using $n+1$ as the starting integer instead of $n$ ) breaks the consecutive integer relationship.

When to use it

Use when a problem gives you a relationship (like a sum or product) involving integers that follow each other with no gaps.

Level

Usually taught in: Algebra I

Worked examples

1. Find two consecutive integers that sum to 21

Problem

Find two consecutive integers whose sum is 21.

$n + (n+1) = 21$
Let $n$ be the first integer; then the next integer is $n+1$ . Their sum is 21.
$2n + 1 = 21$
Combine like terms: $n + n = 2n$ , and the constant term is 1.
$2n = 20$
Subtract 1 from both sides to isolate the term with the variable.
$n = 10$
Divide both sides by 2 to solve for $n$ .

Answer: $n = 10; \text{ the consecutive integers are } 10 \text{ and } 11$

We translate 'two consecutive integers' into $n$ and $n+1$ , then write their sum as an equation. After combining like terms and using inverse operations, we find $n = 10$ . Since $n$ is the first integer, the second is $n+1 = 11$ . Verification: $10 + 11 = 21$ .

2. Find three consecutive integers with a sum of -15

Problem

The sum of three consecutive integers is

-15

. Find them.

$n + (n+1) + (n+2) = -15$
Let $n$ be the first integer; the second and third are $n+1$ and $n+2$ . All three sum to $-15$ .
$3n + 3 = -15$
Combine like terms: three copies of $n$ give $3n$ , and the constants $0 + 1 + 2 = 3$ .
$3n = -18$
Subtract 3 from both sides to isolate the term containing $n$ .
$n = -6$
Divide both sides by 3.

Answer: $n = -6; \text{ the consecutive integers are } -6, -5, \text{ and } -4$

This problem shows that consecutive integers can include negative numbers. The algebra is identical to positive cases, but the final answer is negative. After solving for $n = -6$ , we find the other integers by adding 1 and 2: the set is $-6, -5, -4$ . Verification: $-6 + (-5) + (-4) = -15$ .

3. Game points across consecutive levels

Problem

A game awards points equal to the level number for each level completed. A player earns 42 points total by completing three consecutive levels. What are the three level numbers?

$n + (n+1) + (n+2) = 42$
Let $n$ be the first level number. Each level awards points equal to its number, so the total is $n + (n+1) + (n+2)$ , which equals 42.
$3n + 3 = 42$
Combine like terms: $n + n + n = 3n$ , and $1 + 2 = 3$ .
$3n = 39$
Subtract 3 from both sides.
$n = 13$
Divide both sides by 3.

Answer: $n = 13; \text{ the level numbers are } 13, 14, \text{ and } 15$

In word problems, the variable represents a real-world quantity—here, a level number. The setup is the same: three consecutive integers summing to 42. After solving, we find the levels are 13, 14, and 15. Verification: $13 + 14 + 15 = 42$ points awarded.

Common mistakes

Where Consecutive Integer Problems usually goes wrong

The mistake	Why it is wrong	The fix
Writing $n = 8$ as your final answer when the problem asks for the consecutive integers.	The variable $n$ represents only the first integer; you must compute all of them by substituting $n$ into each expression like $n+1$ and $n+2$ .	After solving for $n$ , state the complete set: if $n = 8$ and the problem asks for three consecutive integers, write '8, 9, and 10.'
Writing $n + n + 1 + 2 = 25$ instead of $n + (n+1) + (n+2) = 25$ for three consecutive integers.	Without parentheses, ' $n + 1 + 2$ ' becomes just the constants 3; you lose the $n$ terms and solve a completely different equation.	Always write parentheses around each consecutive integer: $n$ , $(n+1)$ , $(n+2)$ , etc., so that the $+1$ and $+2$ clearly belong inside each term.
Using $n, n+2, n+4$ to represent three general consecutive integers.	This spacing represents every other integer (consecutive evens or consecutive odds), not each integer in order; consecutive integers always increase by 1.	Use $n, n+1, n+2$ for all consecutive integers; reserve $n, n+2, n+4$ only when the problem specifically asks for consecutive even or consecutive odd integers.

Tips and when to use something else

Always begin by stating 'Let $n$ be the first integer' (or whichever integer you choose) so that anyone reading your work—including yourself later—knows what $n$ represents.
After solving for $n$ , substitute its value back into every consecutive-integer expression ( $n+1$ , $n+2$ , etc.) to find all integers the problem is asking for.
If the problem specifies consecutive even integers or consecutive odd integers, use $n, n+2, n+4$ instead, and explicitly state that $n$ is even or odd.
If the integers in a problem are related by operations other than 'add 1' (for example, 'the second is twice the first'), use Solving by Substitution instead of the consecutive integer pattern.

Related formulas

Linear EquationSolve for the value of x in any linear equation by rearranging to isolate x, finding the unique solution where the expression equals zero.Two-Step EquationsSolve two-step equations of the form ax + b = c by undoing operations in reverse order to isolate the variable using inverse operations.Distributive PropertyThe Distributive Property lets you multiply a number by a sum or difference by distributing the multiplication to each term separately.Solving by SubstitutionSolving by Substitution solves a system of equations by replacing one variable with its equivalent expression, turning two equations into one.

Frequently asked questions

Can consecutive integers be negative?: Yes. Consecutive integers can start at any integer—positive, negative, or zero. For example, $-5, -4, -3$ are three consecutive integers. The algebraic method is identical whether the integers are positive or negative.
What is the difference between consecutive integers and consecutive even or odd integers?: Consecutive integers ( $n, n+1, n+2, \ldots$ ) include every whole number in a row. Consecutive even integers or consecutive odd integers ( $n, n+2, n+4, \ldots$ ) skip every other number. Use the second pattern only when the problem explicitly says 'even' or 'odd.'
Do I need to substitute back, or is n the answer?: You must substitute back. The variable $n$ represents only the first integer; the problem usually asks for all the integers in the sequence. If $n = 7$ and you need three consecutive integers, your answer is 7, 8, and 9—not just 7.
Can I use a different variable or starting point?: Yes. Some people use $n-1, n, n+1$ (with $n$ in the middle) or other setups. However, using $n$ as the first integer keeps the arithmetic simpler and requires fewer parentheses, making it the standard approach.

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Reviewed 2026-09-18