Use the formula $$ \mathrm{S}=\mathrm{n}^{2} $$ to find the sum of $$ 1+3+5+\cdots+261 $$. (Hint: To find $$ n $$, add 1 to the last term and divide by 2.) $$ 1+3+5+\cdots+261=252 $$ (Simplify your answer.)

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To find the sum of the series $$ 1 + 3 + 5 + \cdots + 261 $$ using the formula $$ \mathrm{S} = \mathrm{n}^2 $$, follow these steps:

1. **Determine the number of terms ($$ n $$)**:
   
   The last term of the series is 261. For a series of consecutive odd numbers, the $$ n $$-th term is given by:
   $$
   \text{Last term} = 2n - 1
   $$
   
   Solve for $$ n $$:
   $$
   2n - 1 = 261 \
   2n = 262 \
   n = 131
   $$
   
2. **Calculate the sum ($$ \mathrm{S} $$)**:
   
   Using the formula $$ \mathrm{S} = \mathrm{n}^2 $$:
   $$
   \mathrm{S} = 131^2 = 17161
   $$
   
Therefore, the sum of the series $$ 1 + 3 + 5 + \cdots + 261 $$ is **17,161**.
The sum of the series $$ 1 + 3 + 5 + \cdots + 261 $$ is **17,161**.

Use the formula \( \mathrm{S}=\mathrm{n}^{2} \) to find the sum of \( 1+3+5+\cdots+261 \). (Hint: To find \( n \), add 1 to the last term and divide by 2.) \( 1+3+5+\cdots+261=252 \) (Simplify your answer.)

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