**Middle School Number Sense Lesson 28: Adding a Sequence of Consecutive Odd Numbers (Starting with 1)**

I've said before that sometimes the math is so slick, it almost feels like cheating. Today's topic is another one of those.... This showed up

**18 times**this year in middle school--that's an average of

**once per test**. It was all over the place--as early as question # 7 and as late as question # 79, with the median at

**question # 45**.

**Number Dojo Level: 135**

This question will appear in several forms:

- 1 + 3 + 5 + 7 + ... + 19 = ___
- The sum of the first 10 odd positive integers is ___
- 19 + 17 + 13 + ... + 3 + 1 = ___
- (1 + 3 + 5 + ... + 79) ÷ (1 + 3 + 5 + 7) = ___

The answer to each of the above examples is

**100**.

**How to Solve:**

**Square the # of terms.**- You're done. Really.

**Tips:**

- Make sure you're looking at a sequence that starts with one and doesn't skip any odd numbers.
- To identify the # of terms, add 1 to the highest number, and then divide by 2.
- Use your various squaring skills (like SquareEnd5, Square11-20, and Square21-30) to expedite these calculations.

**Example 1: 1 + 3 + 5 + 7 + ... + 31 = ___**

- Figure out how many terms are in the sequence. (31 + 1) ÷ 2 = 32 ÷ 2 =
**16**. - Square 16 to get
**256**.

**Example 2: 1 + 3 + 5 + ... + 49 = ___**

- Figure out how many terms are in the sequence. (49 + 1) ÷ 2 =
**25**. - Square 25 and get
**625**.

**Example 3: The sum of the first 50 odd whole numbers is ___**

- These are my favorite versions of this problem. There are 50 terms (as stated above).
- Square 50 to get
**2500**.

**Example 4: 25 + 23 + 21 + ... + 3 + 1 = ___**

- Figure out how many terms: (25 + 1) ÷ 2 =
**13**. - Square 13 to get
**169**.

**Example 5: 0.1 + 0.3 + 0.5 + ... + 1.9 + 2.1 = ___**

- Figure out how many terms (ignore the decimals for now). (21 + 1) ÷ 2 =
**11**. - Square 11 to get
**121**. Remember there is one decimal place though...the answer is**12.1**.

**Example 6: (1 + 3 + 5 + ... + 39) ÷ (1 + 3 + 5 + ... + 9) = ___**

- This isn't as hard as it looks. Figure out the sums first, and then divide.
- The first sequence has (39 + 1) ÷ 2 =
**20**terms, so the sum is 20 x 20 =**400**. - The second sequence has (9 + 1) ÷ 2 =
**5**terms, so the sum is 5 x 5 =**25**. - 400 ÷ 25 =
**16**. You could have also divided the # of terms and squared it. (20 ÷ 5) =**4**. 4 x 4 =**16**.

**Here's a free worksheet to help you practice AddSeqOdd:**

addseqodd.pdf |

**Up Next for Middle School: OrderOfOperF**