Consecutive Integers Calculator

Find consecutive, even, or odd integers that sum to a given value with step-by-step solutions.

Select Type & Enter Values

Result

Consecutive Integers
6, 7, 8, 9, 10
sum = 40
Type Consecutive
Count 5
First Integer 6
Last Integer 10
Verified Sum 40

Step-by-Step Solution

S = n/2 x (first + last)

Understanding Consecutive Integers

Consecutive integers are integers that follow each other in order without any gaps. For example, 4, 5, 6, 7 are four consecutive integers. Each integer is exactly 1 more than the previous one. Problems involving consecutive integers are common in algebra and number theory.

The concept extends to consecutive even integers (e.g., 2, 4, 6, 8) and consecutive odd integers (e.g., 1, 3, 5, 7), where each number differs from the previous by 2 instead of 1.

Formulas for Consecutive Integers

Consecutive Integers

n integers starting at a: a, a+1, a+2, ..., a+(n-1)

S = n*a + n(n-1)/2

Consecutive Even Integers

n even integers starting at a: a, a+2, a+4, ..., a+2(n-1)

S = n*a + n(n-1)

Consecutive Odd Integers

n odd integers starting at a: a, a+2, a+4, ..., a+2(n-1)

S = n*a + n(n-1)

Arithmetic Series Sum

General formula for a sequence with first term a and last term l.

S = n/2 x (a + l)

Finding the First Term

Solve for the starting integer a given the sum S and count n.

a = (S - n(n-1)/2) / n

Average of Sequence

The average of consecutive integers equals the middle value.

avg = S / n

How to Find Consecutive Integers with a Given Sum

To find n consecutive integers that sum to S, represent the integers as a, a+1, a+2, ..., a+(n-1). Their sum is:

S = n*a + (0 + 1 + 2 + ... + (n-1)) = n*a + n(n-1)/2

Solving for a: a = (S - n(n-1)/2) / n. If a is an integer, the solution exists. If a is not an integer, then no set of n consecutive integers sums to S.

Example: Find 3 Consecutive Integers that Sum to 27

Using the formula: a = (27 - 3(2)/2) / 3 = (27 - 3) / 3 = 24 / 3 = 8. So the integers are 8, 9, and 10. Check: 8 + 9 + 10 = 27.

Even and Odd Consecutive Integers

For consecutive even integers, the step size is 2 instead of 1. The integers are a, a+2, a+4, ..., a+2(n-1). The sum formula becomes S = n*a + n(n-1). Solving: a = (S - n(n-1)) / n. For a valid solution, a must be an even integer.

Consecutive odd integers follow the same pattern with step size 2, but the starting value a must be odd.

Practical Applications

  • Algebra word problems: Many textbook problems ask students to find consecutive integers with certain properties.
  • Number theory: Properties of consecutive integer sums relate to divisibility and factorization.
  • Puzzles and competitions: Math competitions frequently feature consecutive integer problems.
  • Scheduling: Consecutive numbering is used in seat assignments, page numbering, and inventory systems.

Interesting Properties

  • The sum of any two consecutive integers is always odd.
  • The product of n consecutive integers is always divisible by n!.
  • Every integer greater than 0 can be expressed as a sum of consecutive positive integers (except powers of 2).
  • The average of consecutive integers is always the middle number (or the average of the two middle numbers if n is even).