Consecutive Odd Numbers: Definition, Formula, Sum, Solved Examples & Free Worksheet

Consecutive odd numbers are odd numbers that come one right after another, always 2 apart. Think of 7, 9, 11, 13. If you have ever been asked to “find three consecutive odd numbers whose sum is 51,” you have already met them. They turn up in school algebra, number series, and aptitude tests, and the good news is that one simple idea explains almost every question. In this guide, I will walk you through the definition, the formulas, the sum and average, and a set of solved examples. A free worksheet waits at the end.

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What Are Consecutive Odd Numbers?

An odd number is any whole number that leaves a remainder of 1 when you divide it by 2. So 1, 3, 5, 7 and 9 are odd. Now pick odd numbers that sit side by side in that list, with none skipped, and you have consecutive odd numbers.

For example, 15, 17, 19 works. But 15, 19, 23 do not, because 17 and 21 are missing.

A few things are worth remembering:

The gap between neighbours is always 2.
Every number ends in 1, 3, 5, 7 or 9.
Negative numbers count too, such as -5, -3, -1, 1.
Multiply any set of them and the answer is odd.

Students often confuse these with ordinary consecutive numbers like 4, 5, 6, where the gap is 1. If you keep “gap of 2” in your head, you will avoid most errors.

Consecutive Odd Numbers in Terms of x

In algebra we don’t know the numbers yet, so we give the first one a letter. When you write consecutive odd numbers in terms of x, you let x be an odd number and add 2 each time to get the next one.

How many numbers	In terms of x
2	x, x + 2
3	x, x + 2, x + 4
4	x, x + 2, x + 4, x + 6
5	x, x + 2, x + 4, x + 6, x + 8

Look closely at the pattern: 2, 4, 6, 8. Not 1, 2, 3, 4. In my classes, this is the slip I see most. Adding 1 gives an even number, and the whole set falls apart.

You can also build the set around the middle. For three numbers, write x − 2, x, x + 2. This is handy when a question gives you the average or the middle value.

Consecutive Odd Numbers in Terms of n

Some books prefer n to x. Here n can be any integer, and the expression 2n + 1 guarantees an odd result every time.

How many numbers	In terms of n
2	2n + 1, 2n + 3
3	2n + 1, 2n + 3, 2n + 5
4	2n + 1, 2n + 3, 2n + 5, 2n + 7

You may also see 2n − 1, 2n + 1, 2n + 3. That is just as correct. Try n = 4 in the first set: 2(4) + 1 = 9, then 11, then 13. Three consecutive odd numbers.

The only real difference between the two methods is this. With x, you have to say that x is odd. With n, the “2n” part takes care of that for you.

Consecutive Odd Numbers Formula

All the formulas below come from one fact: the numbers form a pattern that grows by 2 each step.

The nth odd number: 2n − 1. The 10th odd number is 2(10) − 1 = 19.
Sum of the first n odd numbers: n². Check it: 1 + 3 + 5 + 7 + 9 = 25, and 5² = 25.
How many odd numbers lie from a to b: (b − a) ÷ 2 + 1.
Sum from a to b: (a + b) ÷ 2 × number of terms.
Sum of k consecutive odd numbers starting at x: kx + k(k − 1).

Let’s test formula 5 with three numbers. It gives 3x + 3(2) = 3x + 6, and if you add x + (x + 2) + (x + 4) by hand, you get the same thing.

Some patterns worth knowing:

Add two consecutive odd numbers and the result is even and divisible by 4. For example, 7 + 9 = 16.
Add three and the result is divisible by 3. For example, 7 + 9 + 11 = 27.
An even count of them always adds up to an even number, and an odd count adds up to an odd number.

Sum of Consecutive Odd Numbers

There are two cases here. Either you start from 1, or you don’t.

Starting from 1: just square the count. The first 8 odd numbers add up to 8² = 64. Verify: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64.

Starting somewhere else: use the formula for evenly spaced numbers. Suppose you need 21 + 23 + 25 + … + 49.

Number of terms: (49 − 21) ÷ 2 + 1 = 15
Sum: (21 + 49) ÷ 2 × 15 = 35 × 15 = 525

Consecutive Odd Numbers List

Here are the first twenty, which are good to know by heart:

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39

And a few groups of three to show you can start anywhere:

1, 3, 5
11, 13, 15
21, 23, 25
101, 103, 105

The only rule is that the first number must be odd, and you keep adding 2.

Consecutive Odd Numbers 1 to 100

Between 1 and 100 there are exactly 50 odd numbers. I have split them into rows of ten so they are easy to scan.

Row	Numbers	Row sum
1	1, 3, 5, 7, 9, 11, 13, 15, 17, 19	100
2	21, 23, 25, 27, 29, 31, 33, 35, 37, 39	300
3	41, 43, 45, 47, 49, 51, 53, 55, 57, 59	500
4	61, 63, 65, 67, 69, 71, 73, 75, 77, 79	700
5	81, 83, 85, 87, 89, 91, 93, 95, 97, 99	900

The row sums add up to 2,500. That matches n², since 50² = 2,500. It is a nice way to see the formula working on real numbers.

Consecutive Odd Numbers Average

Because the numbers are evenly spaced, finding the average is quick.

Average = (first number + last number) ÷ 2

There is an even faster trick:

With an odd count (3, 5, 7 numbers), the average is the middle number.
With an even count (4, 6 numbers), the average is the mean of the two middle numbers.

Take 13, 15, 17, 19, 21. The middle number is 17, and (13 + 21) ÷ 2 is 17 as well.

For all the odd numbers from 1 to 99, the average is (1 + 99) ÷ 2 = 50. Or divide the total by the count: 2,500 ÷ 50 = 50.

Solved Examples

Example 1: Three consecutive odd numbers add up to 51. Find them.

Let the numbers be x, x + 2, x + 4.
3x + 6 = 51
3x = 45
x = 15

The numbers are 15, 17, 19. Check: 15 + 17 + 19 = 51.

Example 2: Four consecutive odd numbers add up to 80. Find them.

Let them be x, x + 2, x + 4, x + 6.
4x + 12 = 80
4x = 68
x = 17

The numbers are 17, 19, 21, 23.

Example 3: The product of two consecutive odd numbers is 143. Find them.

Let the numbers be x and x + 2.
x(x + 2) = 143
x² + 2x − 143 = 0
(x + 13)(x − 11) = 0
x = 11 or x = −13

So the pair is 11 and 13. The negative pair −13 and −11 also works, since (−13)(−11) = 143.

Example 4: Find the sum of the first 15 odd numbers.

Sum = 15² = 225.

Example 5: Write three consecutive odd numbers in terms of n, then find them if their sum is 45.

The numbers are 2n + 1, 2n + 3, 2n + 5. Their sum is 6n + 9.
6n + 9 = 45, so n = 6.

The numbers are 13, 15, 17.

Example 6: The average of five consecutive odd numbers is 41. Find them.

With five numbers, the average is the middle one. So the middle number is 41, and the set is 37, 39, 41, 43, 45.

Common Mistakes to Avoid

Adding 1 instead of 2. x, x + 1, x + 2 gives consecutive numbers, not consecutive odd numbers.
Starting with an even x. The whole set then turns out even.
Dropping the negative answer. Equations like x(x + 2) = 143 usually have two valid pairs.
Treating “consecutive” and “odd” as separate ideas. Consecutive means no odd number is skipped.
Using n² from any starting point. That shortcut only works when you begin at 1.

Free Worksheet: Consecutive Odd Numbers

Give these a go before you look at the answers.
1. Write the next three consecutive odd numbers after 37.
2. Find three consecutive odd numbers whose sum is 75.
3. What is the sum of the first 20 odd numbers?
4. Find four consecutive odd numbers whose sum is 112.
5. Find the average of five consecutive odd numbers starting from 31.
6. Find the sum of all odd numbers from 51 to 99.
7. The product of two consecutive odd numbers is 255. Find them.
8. The largest of three consecutive odd numbers is 41. Find their sum.
9. Write three consecutive odd numbers in terms of n, starting with 2n + 1.
10. True or false: 9 and 13 are consecutive odd numbers.
Answers
1. 39, 41, 43
2. 23, 25, 27 (3x + 6 = 75, so x = 23)
3. 400 (20²)
4. 25, 27, 29, 31 (4x + 12 = 112, so x = 25)
5. 35 (the numbers are 31, 33, 35, 37, 39)
6. 1,875 (25 terms with an average of 75)
7. 15 and 17, or −17 and −15
8. 117 (the numbers are 37, 39, 41)
9. 2n + 1, 2n + 3, 2n + 5
10. False, because 11 sits between them

Conclusion

Everything about consecutive odd numbers comes back to the gap of 2. Write them as x, x + 2, x + 4 or as 2n + 1, 2n + 3, 2n + 5, use n² when you start from 1, and look for the middle number when you need an average. Work through the worksheet a couple of times and these questions will start to feel routine.

Frequently Asked Questions

What are consecutive odd numbers?

They are odd numbers that follow one another with a difference of 2, like 21, 23, 25.

How do I write them in algebra?

Use x, x + 2, x + 4 (with x odd), or 2n + 1, 2n + 3, 2n + 5 (with n any integer).

What is the sum of the first n odd numbers?

It is n². The first 10 odd numbers add up to 100.

How many odd numbers are there from 1 to 100?

Fifty.

Is the sum of two consecutive odd numbers odd or even?

Even, and always divisible by 4.