Consecutive Integers Calculator
Find a run of consecutive integers from a sum, generate a sequence from a starting number, or check consecutive even and odd integer problems step by step.
Consecutive integer sequence solver
A consecutive integers calculator finds whole numbers that follow each other in order, such as 7, 8, 9, 10. It can also work with consecutive even integers, such as 12, 14, 16, and consecutive odd integers, such as 5, 7, 9.
A consecutive integers calculator finds a sequence of whole numbers that follow each other without gaps. Consecutive integers increase or decrease by 1. For example, 4, 5, 6, and 7 are consecutive integers. These calculators solve equations, sums, averages, and number sequence problems used in algebra and arithmetic.
Use the sum mode when a word problem says "the sum of five consecutive integers is 80." Use the generate mode when you already know the first integer and want the next values, total, average, and equation.
Use the product mode for problems such as "two consecutive odd integers have a product of 63." The calculator checks whether a valid whole-number sequence exists, and if the requested sum or product cannot produce consecutive integers of the selected type, it explains why instead of forcing a decimal result.
Consecutive integer result
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Sequence
All matching sequences
Sum
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Total of all terms
Average
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Middle value for symmetric sequences
First and last
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Equation
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Arithmetic sequence formula
Step-by-step solution
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How to use the consecutive integers calculator
- Choose the mode: Find integers from a sum, find integers from a product, or generate a sequence from a known first integer.
- Select the sequence type: Use consecutive integers for a step of 1, or choose even or odd integers for a step of 2.
- Enter the count: Type how many integers should be in the sequence.
- Enter the sum, product, or first integer: The input label changes automatically based on the selected mode.
- Read the result: The calculator lists the sequence, total sum or product, average, first term, last term, and the equation used.
Consecutive integers formula
Consecutive integers form an arithmetic sequence. The first term is a, the number of terms is n, and the step is d. Standard consecutive integers use d = 1, while consecutive even and odd integers use d = 2.
Sum = n / 2 × (first + last)
Last = first + (n - 1)d
First = Sum / n - d(n - 1) / 2
The average of the sequence is always the midpoint between the first and last terms. That is why a sum problem can be solved by finding the average first, then spreading the integers evenly around it.
Formula reference: Wolfram MathWorld - Arithmetic Progression.
When does a sum have a whole-number solution?
Not every sum can create the requested sequence. A valid answer exists only when the formula gives a whole-number first term and, for even or odd sequences, the first term has the right parity. Use these checks to spot impossible inputs before doing the full calculation.
| Sequence type | Step | Quick test | Example that works |
|---|---|---|---|
| Regular consecutive integers | 1 | S / n - (n - 1) / 2 must be an integer. | S = 80, n = 5 gives first = 14. |
| Consecutive even integers | 2 | S / n - (n - 1) must be an even integer. | S = 60, n = 4 gives first = 12. |
| Consecutive odd integers | 2 | S / n - (n - 1) must be an odd integer. | S = 45, n = 3 gives first = 13. |
| Odd count shortcut | 1 or 2 | If n is odd, S / n is the middle value. | S = 75, n = 5 gives middle = 15. |
If the quick test fails, the calculator will return an explanation instead of inventing a decimal "integer." This is especially useful for homework where the final answer must be a whole-number sequence.
Integer reference: Encyclopaedia Britannica - Integer.
Product of consecutive integers
Product problems behave differently from sum problems. A sum leads to a linear equation, but a product creates a polynomial such as x(x + 1) = P or x(x + 1)(x + 2) = P. This calculator handles common product problems by searching whole-number sequences and returning every matching sequence it finds.
Two integers
For two regular consecutive integers, use x(x + 1) = P. Example: product 56 gives 7 and 8, and also -8 and -7.
Odd or even products
For consecutive odd or even integers, the step is 2, so the model is x(x + 2) = P. Example: odd product 63 gives -9, -7 and 7, 9.
Zero products
A product is zero whenever one term is zero. That means several standard or even sequences can match the same product of 0.
Translate word problems into equations
Consecutive integer problems are often more about translation than arithmetic. First choose a variable for the smallest number, then write the rest of the sequence with the correct step.
| Problem phrase | Use this setup | Equation pattern | What to enter |
|---|---|---|---|
| "Three consecutive integers have a sum of 42" | x, x + 1, x + 2 | x + (x + 1) + (x + 2) = 42 | Sum mode, count 3, standard type, sum 42 |
| "Four consecutive even integers total 100" | x, x + 2, x + 4, x + 6 | x + (x + 2) + (x + 4) + (x + 6) = 100 | Sum mode, count 4, even type, sum 100 |
| "Two consecutive odd integers have a product of 63" | x, x + 2 | x(x + 2) = 63 | Product mode, count 2, odd type, product 63 |
| "The first of five consecutive odd integers is 9" | 9, 11, 13, 15, 17 | a + kd, where a = 9 and d = 2 | Generate mode, count 5, odd type, first 9 |
| "Find a range of six integers starting at -3" | -3, -2, -1, 0, 1, 2 | a + kd, where a = -3 and d = 1 | Generate mode, count 6, standard type, first -3 |
Equation-solving reference: Paul's Online Math Notes, Lamar University - Applications of Linear Equations.
Decreasing sequences, negative values, and answer order
Most algebra books list consecutive integers from smallest to largest, but a real problem may describe them in reverse order or include negative values. The same sequence can be written either direction as long as every number keeps the correct step.
Decreasing by 1
If a problem says the numbers decrease, solve the increasing sequence first and reverse the output. The set 4, 5, 6, 7 can also be written 7, 6, 5, 4.
Crossing zero
A sequence can move from negative to positive values, such as -2, -1, 0, 1, 2. Zero is an integer, so it belongs in the answer when the range crosses it.
Smallest vs. largest
If the question asks for the smallest number, use the first output value. If it asks for the largest number, use the last output value.
Worked examples
| Problem | Setup | Sequence | Check |
|---|---|---|---|
| Five consecutive integers sum to 80 | n = 5, S = 80, d = 1 | 14, 15, 16, 17, 18 | 14 + 15 + 16 + 17 + 18 = 80 |
| Four consecutive even integers sum to 60 | n = 4, S = 60, d = 2 | 12, 14, 16, 18 | 12 + 14 + 16 + 18 = 60 |
| Three consecutive odd integers sum to 45 | n = 3, S = 45, d = 2 | 13, 15, 17 | 13 + 15 + 17 = 45 |
| Start at -2 and list six integers | a = -2, n = 6, d = 1 | -2, -1, 0, 1, 2, 3 | Sum = 3, average = 0.5 |
What to check in word problems
Consecutive integer word problems often hide the sequence type in a short phrase. Read the wording carefully before choosing the calculator setting.
"Consecutive integers"
Use a step of 1. Examples include 6, 7, 8 and -1, 0, 1.
"Consecutive even integers"
Use even terms and a step of 2. Examples include 10, 12, 14.
"Consecutive odd integers"
Use odd terms and a step of 2. Examples include 11, 13, 15.
Interesting fact
The sum of any odd number of consecutive integers is always divisible by the number of terms. For example, five consecutive integers always have a middle integer, and the sum is five times that middle value. This is why many textbook problems use 3, 5, or 7 consecutive integers: the middle-number method makes the algebra especially clean.
Frequently Asked Questions
What are consecutive integers in a number sequence?
Consecutive integers are whole numbers that follow one another with no gaps in the sequence. The range can include negative integers, zero, and positive integers, such as -2, -1, 0, 1, 2. In algebra, the first value is often written as a variable like x, then the next values become x + 1, x + 2, and so on.
How does the calculator find consecutive integers from a sum?
The calculator uses your input sum and the number of integers to find the average, then places the terms evenly around that average. For a formula-based solution, use first = Sum / n - d(n - 1) / 2, where d is the step. The output gives the full sequence, the equation used, and an explanation you can check against the original math problem.
Why does the calculator sometimes output no integer answer?
The calculated first term may be a decimal, or the value may not match the selected even or odd requirement. For example, three consecutive integers cannot have a sum of 10 because the average would be 10 / 3, not a whole-number middle term. In that case, the calculator reports that no valid solution exists because consecutive integer problems normally require integer answers.
Can the calculator solve product problems too?
Yes. Product mode searches for consecutive integer sequences whose terms multiply to the target product. This is useful for algebra problems such as x(x + 1) = 56 or x(x + 2) = 63. If more than one sequence works, such as a positive pair and a negative pair, the calculator lists all matching outputs.
How are consecutive even and odd integers handled in the equation?
Regular consecutive integers increase by 1, while consecutive even and consecutive odd integers use a step of 2 in the formula. Even sequences must start with an even number, and odd sequences must start with an odd number. The calculator checks this rule automatically so the final answer matches the selected sequence type.
Can the number range include zero or negative integers?
Yes. Integers include negative whole numbers, zero, and positive whole numbers. A valid consecutive integer sequence can cross zero, such as -3, -2, -1, 0, 1, as long as each term follows the selected step. This matters in algebra because a variable can represent a negative starting value just as easily as a positive one.
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Disclaimer: This consecutive integers calculator is for general educational and informational use only. It is designed to help with arithmetic sequence practice, algebra homework, and number-pattern checks. It may not match every classroom convention, teacher preference, textbook notation, or assessment format. Always review the required answer form, show your own work when required, and confirm whether the problem asks for regular consecutive integers, consecutive even integers, or consecutive odd integers.
Last updated: May 19, 2026