Built and fact-checked by the DocNectar team — see our editorial standards
Key Features
Instant Calculation
Get accurate results in real time with our optimized algorithm.
Mobile Friendly
Fully responsive design. Works on all devices & screen sizes.
Privacy Focused
Your data stays on your device. We don't store any inputs.
100% Free
No hidden costs. This tool is completely free forever.
Consecutive integer problems are a classic staple of introductory algebra: "the sum of three consecutive integers is 24 — what are the three numbers?" These problems are popular in textbooks because they're a clean, concrete way to practice translating a word problem into an algebraic equation — letting the first integer be some unknown n, then expressing the rest as n+1, n+2, and so on, before solving for n. This tool solves that classic direction — finding the starting integer (and the full sequence) given a target sum and a count of integers — or runs the calculation in reverse, listing a sequence and computing its sum given just a starting integer and how many consecutive integers to include.
How it works
The sum of n consecutive integers starting at s follows a direct formula: n×s + n(n−1)/2 — this comes from adding s + (s+1) + (s+2) + ... up to n terms, and simplifying using the standard sum-of-consecutive-numbers pattern. Given a target sum and count, the tool solves this formula algebraically for s. Given a starting integer and count instead, the tool simply plugs both values into the formula directly to compute the sum, then lists out the full sequence of integers involved.
- Enter given.
- Enter sum (if mode is "from sum").
- Enter starting integer (if mode is "from start").
- Enter count of consecutive integers.
- Click Calculate to see your results.
Examples
Solving for the start
Three consecutive integers summing to 24: solving gives a starting integer of 7, so the sequence is 7, 8, 9 (7+8+9 = 24).
Solving for a longer sequence
Five consecutive integers summing to 100: solving gives a starting integer of 18, so the sequence is 18, 19, 20, 21, 22 (which indeed sums to 100).
Working forward from a known start
Starting at 12 with a count of 4 consecutive integers gives the sequence 12, 13, 14, 15, with a sum of 54 — the reverse direction, useful when you already know where a sequence begins.
Who should use it
- Solving classic algebra word problems about consecutive integers.
- Quickly generating a sequence and its sum for a known starting point.
- Checking homework answers for consecutive integer problems.
Industry applications
- Education (algebra coursework)
- Puzzle and recreational mathematics
Advantages
- Solves in both directions: sum-and-count to sequence, or start-and-count to sum.
- Lists the full resulting sequence, not just the starting value.
- Clearly flags when no integer solution exists.
Limitations
- Only handles consecutive integers (step of 1) — not consecutive even/odd numbers or other fixed-step sequences.
Common mistakes to avoid
- Assuming every sum is achievable with any chosen count of consecutive integers — some combinations simply have no integer solution.
- Setting up the word problem's equation incorrectly, such as forgetting to add all n terms or mislabeling the count of integers involved.
- Confusing consecutive integers (step of 1) with consecutive even or odd integers (step of 2), which need a different formula.
Best practices
- When solving consecutive integer word problems by hand, set up the algebraic sum formula first rather than guessing and checking.
- Double check your answer by adding the resulting sequence back together to confirm it matches the target sum.
- For even/odd consecutive integer problems, adjust your formula's step size before relying on standard consecutive-integer logic.
Tips
- For consecutive even or odd integers (step of 2), adjust the count-related terms in the formula accordingly — this tool is specifically for step-1 consecutive integers.
- Always verify your final sequence by adding it up manually as a sanity check.