Use the formula S​ = 

n squaredn2 

to find the sum of 1​ + 3​ + 5​ + ...​ + 

883883. 

​(Hint: To find​ n, add 1 to the last term and divide by​ 2.)

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Part 1

1​ + 3​ + 5​ + ...​ + 883

​= 

To find the sum of the series \(1 + 3 + 5 + \dots + 883\), we can again use the formula for the sum of the first \(n\) odd numbers:

\[
S = n^2
\]

### Step 1: Find \(n\)
The hint suggests that to find \(n\), we need to add 1 to the last term (883) and divide by 2:

\[
n = \frac{883 + 1}{2} = \frac{884}{2} = 442
\]

So, \(n = 442\).

### Step 2: Calculate the sum \(S\)
Now, use the formula \(S = n^2\) to find the sum:

\[
S = 442^2
\]

I'll calculate that next.The sum of the series \(1 + 3 + 5 + \dots + 883\) is \(195364\).

Would you like more details or have any questions? Here are some related questions you might find helpful:

1. How would the sum change if the last term were 885 instead of 883?
2. How does the pattern of odd numbers relate to perfect squares?
3. Can you derive the formula for the sum of odd numbers from first principles?
4. How would you calculate the sum of even numbers up to a specific term, like 882?
5. What is the sum of the first \(k\) even numbers, and how does that compare to the sum of the first \(k\) odd numbers?

**Tip:** For large series, always double-check the value of \(n\) by verifying that \(n \times 2 - 1\) gives the correct last term in the sequence.

Learn how to calculate the sum of the series 1 + 3 + 5 + ... + 883 using the formula S = n^2. This problem demonstrates the arithmetic series and involves finding the sum of odd numbers up to a specific term. Suitable for Grades 7-9 students studying arithmetic sequences and series.

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