Calculate consecutive integers

Calculate consecutive integers that sum to a given total

Consecutive Integers Calculator

What are consecutive integers?

Consecutive integers are integers that differ by 1: n, n+1, n+2, .... The goal is to find the starting value when the sum and the count are known.

3 consecutive integers with sum 12648
Enter parameters
Total sum of all numbers
Number of consecutive integers
Calculation result
First number:
Last number:
Calculation: x = (S - n(n-1)/2) / n
Sequence
Sequence will be displayed here

Consecutive Sequences Info

Properties

Consecutive integers: n, n+1, n+2, n+3, ...

consecutive Δ = 1 arithmetic

Condition: Sum and count must be compatible
Formula: x = (S - n(n-1)/2) / n

Quick examples
3 numbers, sum 15: 4, 5, 6
4 numbers, sum 22: 4, 5, 6, 7
5 numbers, sum 0: -2, -1, 0, 1, 2

Formulas for consecutive integers

Basic formula
\[x = \frac{S - \frac{n(n-1)}{2}}{n}\]

x = first number, S = sum, n = count

Sum formula
\[S = n \cdot x + \frac{n(n-1)}{2}\]

Compute sum from first number and count

Arithmetic series
\[S = \frac{n}{2}(2a + (n-1)d)\] \[\text{with } d = 1\]

General formula with difference d = 1

Simplified form
\[S = \frac{n(x_{first} + x_{last})}{2}\]

Sum as average × count

Last number
\[x_{last} = x_{first} + (n-1)\]

Compute the last number of the sequence

Existence condition
\[x = \frac{S}{n} - \frac{n-1}{2} \in \mathbb{Z}\]

First number must be integer

Example calculations for consecutive integers

Example: 3 consecutive integers with sum 15
Given
Sum S = 15 Count n = 3

Find: 3 consecutive integers that sum to 15

Step 1: Insert formula
x = (S - n(n-1)/2) / n
x = (15 - 3(3-1)/2) / 3
x = (15 - 3×2/2) / 3 = (15 - 3) / 3 = 12 / 3 = 4
Step 2: Build sequence
First number: x = 4
Second number: x + 1 = 5
Third number: x + 2 = 6
Sequence: 4, 5, 6
Step 3: Verification
Sum check: 4 + 5 + 6 = 15 ✓
\[x = \frac{15 - \frac{3(3-1)}{2}}{3} = \frac{15 - 3}{3} = 4\] \[\text{Sequence: } 4, 5, 6\]
Example 2: Impossible case
Sum S = 16 Count n = 3
x = (16 - 3×2/2) / 3 = (16 - 3) / 3 = 13/3 = 4,333...
Problem: x is not integer → No solution possible

Not every combination of sum and count yields integer solutions.

More examples
4 numbers, S = 22:
x = (22-6)/4 = 4 → 4,5,6,7
5 numbers, S = 0:
x = (0-10)/5 = -2 → -2,-1,0,1,2
2 numbers, S = 11:
x = (11-1)/2 = 5 → 5,6
Check solvability
Condition: x must be integer
Odd n: S arbitrary
Even n: S must have specific form
Test: (S - n(n-1)/2) divisible by n?
Properties of consecutive integers
Arithmetic
Difference d = 1
Consecutive
No gaps
Integer
All numbers ∈ ℤ
Linear
Uniform increment

Consecutive integers form the simplest arithmetic sequences

Applications of consecutive integers

Consecutive integers have many applications in mathematics and everyday life:

Puzzles & Games
  • Number puzzles and math games
  • Sudoku and logic puzzles
  • Sequence-based brainteasers
  • Competition math problems
Number theory
  • Sum formulas and arithmetic series
  • Parity properties (even/odd)
  • Divisibility rules and congruences
  • Induction proofs and recurrences
Practical applications
  • Seating and numbering
  • Time intervals and scheduling
  • Product series and inventory numbers
  • Sports events and rankings
Computer Science & Programming
  • Array indexing and loops
  • Algorithm analysis and complexity
  • Data structures and sorting
  • Recursive problem solving

Consecutive integers: Cornerstone of arithmetic series

Consecutive integers are the simplest form of arithmetic sequences and form a basic building block of elementary number theory. The property that neighboring numbers differ by exactly 1 makes them an ideal entry point for understanding patterns and regularities in mathematics. From simple puzzles to complex proofs in number theory – consecutive integers reveal deep mathematical structures and form the basis for understanding arithmetic progressions, sum formulas and analytical problem-solving strategies.

Characteristics
  • Constant difference d = 1
  • Integer elements: n ∈ ℤ
  • Gapless sequence
  • Linear ordering on the number line
Mathematical significance
  • Simple arithmetic sequence
  • Basis for sum formulas
  • Models discrete steps
  • Foundation for induction proofs
Solution strategies
  • Formula x = (S - n(n-1)/2)/n
  • Check existence condition
  • Use symmetry around the mean
  • Verify by summation
Summary

Consecutive integers embody mathematical order in its purest form. These seemingly simple sequences are keys to understanding more complex arithmetic structures. Their applications range from elementary puzzles to rigorous algebraic methods and demonstrate how elementary concepts can lead to deep mathematical insights. From school-level problems to university proofs – consecutive integers remain a fascinating example of mathematical elegance and structure.

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