What is the square root of 1 + 3 + 5 + 7 + .....+ 199? You do not need to add 100 numbers one by one. There is a beautiful pattern: the sum of the first n odd numbers is n². Since 199 is the 100th odd number, the answer is 100.
This clever number pattern is useful for mental math, algebra, number theory, homework, and test preparation. Once you recognize it, a long-looking radical expression becomes a one-step calculation.
The Quick Answer
Why is the answer 100? The sequence starts at 1 and ends at 199. The nth odd number is:
Set the last number equal to 199:
Add 1 and divide by 2:
The sum of the first 100 odd numbers is 100², so:
The Amazing Pattern
Look at the first few sums of consecutive odd numbers:
| Odd Numbers | Sum | Square-Root Result |
|---|---|---|
| 1 + 3 | 4 = 2² | √4 = 2 |
| 1 + 3 + 5 | 9 = 3² | √9 = 3 |
| 1 + 3 + 5 + 7 | 16 = 4² | √16 = 4 |
| 1 + 3 + 5 + 7 + 9 | 25 = 5² | √25 = 5 |
| 1 + 3 + 5 + ... + 199 | 10,000 = 100² | √10,000 = 100 |
The pattern is:
Therefore:
Step-by-Step Solution for the Video
Step 1: Identify the final odd number
The final number in the sequence is 199.
Step 2: Find its position in the odd-number sequence
Use the formula for the nth odd number, 2n − 1:
Step 3: Use the odd-number sum rule
The first 100 odd numbers add up to 100²:
Step 4: Take the square root
Why Does the Pattern Work?
Each new odd number increases the previous square by the next amount needed to create the next square:
Algebraically, the difference between two consecutive squares is:
That difference is exactly the nth odd number. This is why adding consecutive odd numbers creates perfect squares.
A Visual Way to Remember It
Imagine building square shapes with small unit blocks. Start with a 1 × 1 square. To turn it into a 2 × 2 square, add 3 blocks. To turn a 2 × 2 square into a 3 × 3 square, add 5 blocks. The next border needs 7 blocks, then 9 blocks, and so on.
After adding the first n odd numbers, the shape becomes an n × n square. Its area is n², which explains the identity:
More Examples
Use the position of the final odd number to solve these quickly:
| Expression | Number of Odd Terms | Answer |
|---|---|---|
| √(1 + 3 + 5 + 7) | 4 | 4 |
| √(1 + 3 + ... + 19) | 10 | 10 |
| √(1 + 3 + ... + 49) | 25 | 25 |
| √(1 + 3 + ... + 99) | 50 | 50 |
| √(1 + 3 + ... + 199) | 100 | 100 |
Common Mistakes to Avoid
Do not count the last number as the number of terms. For example, 199 is not the 199th odd term. Its position is found using (199 + 1) ÷ 2 = 100.
Do not confuse the sum with the square root of the sum. The sum of the first 100 odd numbers is 10,000, while the square root of that sum is 100.
Also, make sure the sequence starts at 1 and uses consecutive odd numbers. The identity applies to 1 + 3 + 5 + ... without missing terms.
Finally, remember that this shortcut finds the principal square root. The symbol √10,000 means 100. If you solve x² = 10,000, the two solutions are x = 100 and x = −100.
Practice Problems
Try solving these without a calculator:
- √(1 + 3 + 5)
- √(1 + 3 + 5 + 7 + 9 + 11)
- √(1 + 3 + 5 + ... + 29)
- √(1 + 3 + 5 + ... + 79)
- √(1 + 3 + 5 + ... + 299)
Answer Key
- √(1 + 3 + 5) = 3
- √(1 + 3 + 5 + 7 + 9 + 11) = 6
- √(1 + 3 + 5 + ... + 29) = 15
- √(1 + 3 + 5 + ... + 79) = 40
- √(1 + 3 + 5 + ... + 299) = 150
Final Takeaway
The key identity is simple and powerful:
For the video’s challenge, 199 is the 100th odd number:
Instead of performing a long addition, count the odd-number terms, square that count, and then take the square root. This is one of the most useful patterns in elementary number theory and mental math.
Frequently Asked Questions
What is the square root of 1 + 3 + 5 + ... + 199?
The answer is 100, because 199 is the 100th odd number and the sum of the first 100 odd numbers is 100².
What is the formula for the sum of the first n odd numbers?
The formula is 1 + 3 + 5 + ... + (2n − 1) = n².
How do I find the number of terms ending at an odd number?
Use n = (last odd number + 1) ÷ 2. For 199, n = (199 + 1) ÷ 2 = 100.
Is this a square-root trick or an exact rule?
It is an exact mathematical identity for the sum of the first consecutive odd numbers starting at 1. The pattern shown in the video is therefore fully verified by the formula.
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