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๐Ÿ”ข

Consecutive Integers: Sums and Products

Consecutive integers form arithmetic sequences with d=1. Sum = n ร— (first+last)/2. To find integers summing to S: solve for the first term. Even/odd consecutive use d=2.

Concept Fundamentals
n ร— (first+last)/2
Sum formula
1+...+n = n(n+1)/2
Gauss
2, 4, 6... (d=2)
Even
1, 3, 5... (d=2)
Odd

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Gauss summed 1 to 100 as a child: 100ร—101/2 = 5050. Consecutive evens: 2, 4, 6...; odds: 1, 3, 5... Product of n consecutive is divisible by n!.

Key quantities
n ร— (first+last)/2
Sum formula
Key relation
1+...+n = n(n+1)/2
Gauss
Key relation
2, 4, 6... (d=2)
Even
Key relation
1, 3, 5... (d=2)
Odd
Key relation

Ready to run the numbers?

Why: Consecutive integer problems appear in puzzles, algebra, and number theory. Given a target sum, find the sequence. Gauss formula gives 1+2+...+n = n(n+1)/2.

How: For sum of n consecutive: if first = a, sum = nร—a + n(n-1)/2. Set equal to target, solve for a. For product, use factorial or iterative search.

Gauss summed 1 to 100 as a child: 100ร—101/2 = 5050.Consecutive evens: 2, 4, 6...; odds: 1, 3, 5...
Sources:NCTM StandardsKhan Academy Algebra

Run the calculator when you are ready.

Find Consecutive IntegersSum to target, or compute sum/product

Inputs

consecutive.sh
CALCULATED
$ run --mode=findSum
Result
6 set(s)
Sum
15
Product
โ€”
Count
2
7 + 8 = 15
4 + 5 + 6 = 15
1 + 2 + 3 + 4 + 5 = 15
0 + 1 + 2 + 3 + 4 + 5 = 15
-3 + -2 + -1 + 0 + 1 + 2 + 3 + 4 + 5 + 6 = 15
-6 + -5 + -4 + -3 + -2 + -1 + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 15
Consecutive Integers Calculator
7+8 ; 4+5+6 ; 1+2+3+4+5 ; 0+1+2+3+4+5 ; -3+-2+-1+0+1+2+3+4+5+6 ; -6+-5+-4+-3+-2+-1+0+1+2+3+4+5+6+7+8
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Set Sizes

๐Ÿ“ Step-by-Step Breakdown

SETUP
Target sum15
METHOD
Formulasum = n ร— (first + last) / 2
RESULT
Found6 set(s)
Set 17 + 8 = 15
Set 24 + 5 + 6 = 15
Set 31 + 2 + 3 + 4 + 5 = 15
Set 40 + 1 + 2 + 3 + 4 + 5 = 15
Set 5-3 + -2 + -1 + 0 + 1 + 2 + 3 + 4 + 5 + 6 = 15
Set 6-6 + -5 + -4 + -3 + -2 + -1 + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 15

For educational and informational purposes only. Verify with a qualified professional.

๐Ÿงฎ Fascinating Math Facts

๐Ÿ“

Sum of first n naturals = n(n+1)/2 โ€” Gauss formula.

๐Ÿ”ข

Product of n consecutive integers is divisible by n!.

๐Ÿ“‹ Key Takeaways

  • โ€ข Sum of n consecutive integers: n ร— (first + last) / 2
  • โ€ข Powers of 2 cannot be written as sums of consecutive integers
  • โ€ข Consecutive even/odd: same formula with step 2 instead of 1

๐Ÿ’ก Did You Know?

๐Ÿง’Gauss famously summed 1 to 100 as a child using n(n+1)/2 = 5050Source: History
โ†”Consecutive integers alternate odd and evenSource: Number Theory
๐Ÿ”ขAny two consecutive integers are relatively prime (GCD = 1)Source: GCD
๐Ÿ“Product of two consecutive: n(n+1) is never a perfect squareSource: Algebra
๐Ÿ“ŠSum of first n positive integers = n(n+1)/2Source: Gauss Formula
โš Not all numbers can be expressed as sums of consecutive integersSource: Powers of 2

๐Ÿ“– How It Works

For sum = n ร— (first + last) / 2, we solve for first given sum and n. For consecutive integers, last = first + n - 1, so first = (2ร—sum/n - n + 1) / 2. For even/odd consecutive, the step between terms is 2.

๐Ÿ“ Worked Example: 15

Sets summing to 15:

1+2+3+4+5 = 15

4+5+6 = 15

7+8 = 15

๐ŸŽฏ Expert Tips

  • Check if 2ร—sum is divisible by n to quickly filter candidates
  • For odd n, the middle term equals the average (sum/n)
  • Powers of 2 have no representation as sum of consecutive integers

โ“ FAQ

What are consecutive integers?

Whole numbers that follow each other: n, n+1, n+2, ...

Can any number be a sum of consecutive integers?

No. Powers of 2 cannot. All others can.

What is the sum of 1 to n?

n(n+1)/2. For n=100: 5050.

How to find consecutive integers with a given sum?

Solve first = (2ร—sum/n - n + 1)/2 for each n; first must be integer.

What about consecutive even or odd?

Same idea with step 2: first, first+2, first+4, ...

โš ๏ธ Disclaimer: Results are for educational purposes. Verify critical calculations independently.

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