The sum of the first $n$ odd natural numbers is given by the formula:

$S_n = n^2$

Derivation:

1. The sequence of odd natural numbers is: $1, 3, 5, 7, \dots, (2n-1)$.

2. The sum of the first $n$ odd numbers can be expressed as: $S_n = 1 + 3 + 5 + \dots + (2n-1)$

3. The formula for the sum of an arithmetic sequence is: $S = \frac{\text{number of terms}}{2} \times (\text{first term} + \text{last term})$

   • Number of terms = $n$,
   • First term = $1$,
   • Last term = $2n - 1$.

   Substituting: $S_n = \frac{n}{2} \times (1 + (2n-1)) = \frac{n}{2} \times 2n = n^2$

Thus, the sum of the first $n$ odd natural numbers is $n^2$.

Verification:

For $n = 4$: $S_4 = 1 + 3 + 5 + 7 = 16 = 4^2$

Key Insight:

The sum of the first $n$ odd natural numbers directly equals the square of $n$.

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Would you like more examples, a proof by induction, or further clarification?

Related Questions:

1. What is the sum of the first $n$ even natural numbers?
2. Can you derive the sum of odd numbers using mathematical induction?
3. What is the relationship between odd numbers and perfect squares?
4. How does the sum of odd numbers change if the series starts from $3$ instead of $1$?
5. What is the sum of the first $n$ odd natural numbers squared?

Tip:

Odd numbers follow a specific pattern of differences—each increases by 2 compared to the previous. Recognizing this helps in building summation formulas.