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Consecutive Odd Integers — Definition, Formula & Examples

Consecutive odd integers are odd numbers that follow one another in order on the number line, such as 13, 15, 17. Each one is exactly 2 more than the one before it.

A list of integers a1,a2,…,ama_1, a_2, \dots, a_m is a sequence of consecutive odd integers if a1a_1 is odd and ai+1−ai=2a_{i+1} - a_i = 2 for every ii. Equivalently, there is an integer kk such that ai=2k+2i−1a_i = 2k + 2i - 1 for i=1,2,…,mi = 1, 2, \dots, m. Negative odd integers count, so −3,−1,1-3, -1, 1 is a valid list.

Key Formula

2k+1,2k+3,2k+5,…2k+1,\quad 2k+3,\quad 2k+5,\quad \dots
Where:
  • kk = any integer; 2k+1 is then guaranteed to be odd
  • 2k+12k+1 = the first odd integer in the list (in word problems you can simply call it x, where x is odd)

Worked Example

Problem: The sum of three consecutive odd integers is 45. Find the integers.
Step 1: Let x be the smallest odd integer. The next two are each 2 larger, so write an equation for their sum.
x+(x+2)+(x+4)=45x + (x+2) + (x+4) = 45
Step 2: Combine like terms on the left side.
3x+6=453x + 6 = 45
Step 3: Subtract 6 from both sides, then divide by 3. Check that x is odd, which it is.
3x=39⇒x=133x = 39 \quad\Rightarrow\quad x = 13
Answer: The integers are 13, 15, and 17. Check: 13 + 15 + 17 = 45.

Why It Matters

Consecutive odd integer problems are a standard Algebra 1 exercise in turning words into a linear equation. The 2k+1 form also appears in the first proofs you write in geometry or discrete math. For example, (2k+1) + (2k+3) = 4k + 4 shows that the sum of any two consecutive odd integers is a multiple of 4.

Common Mistakes

Mistake: Writing the integers as x, x+1, x+2, or thinking odd numbers need x+1, x+3, x+5.
Correction: Odd integers are 2 apart, so use x, x+2, x+4. Choosing x to be odd is what makes every term odd. The labels x+2 and x+4 are exactly the same ones you use for consecutive even integers.

Related Terms