The sum of the first n odd numbers is n2.
Understanding the Sum of Odd Numbers
Odd numbers follow a pattern: 1, 3, 5, 7, 9, and so on. This is an arithmetic progression (AP) where the first term (a) is 1 and the common difference (d) is 2. We are looking for a formula to directly calculate the sum (Sn) of the first n odd numbers.
Deriving the Formula
We can use the arithmetic series sum formula:
Sn = n/2 * [2a + (n - 1)d]
Where:
- Sn = the sum of the first n terms
- n = the number of terms
- a = the first term (in this case, 1)
- d = the common difference (in this case, 2)
Substituting a = 1 and d = 2 into the formula:
Sn = n/2 [2(1) + (n - 1)2]
Sn = n/2 [2 + 2n - 2]
Sn = n/2 * [2n]
Sn = n2
Therefore, the sum of the first n odd numbers is n2.
Examples
Here's a table demonstrating the sum of the first n odd numbers and the result being n2:
| n (Number of Terms) | Sum of First n Odd Numbers | Calculation | Result (n2) |
|---|---|---|---|
| 1 | 1 | 1 | 1 |
| 2 | 1 + 3 | 4 | 4 |
| 3 | 1 + 3 + 5 | 9 | 9 |
| 4 | 1 + 3 + 5 + 7 | 16 | 16 |
| 5 | 1 + 3 + 5 + 7 + 9 | 25 | 25 |
Conclusion
In summary, the sum of the first n odd numbers is equal to n squared (n2). This is a useful formula to remember when dealing with arithmetic progressions and series.
