To find the sum of the series [tex]\(1 + 3 + 5 + \cdots + 189\)[/tex], we need to follow a few steps. This series is an arithmetic sequence of consecutive odd numbers. Let's break down the solution step by step:
1. Identify the Last Term:
The last term of the series is given as 189.
2. Find the Number of Terms ([tex]\(n\)[/tex]):
The nth term of the series of odd numbers can be expressed as [tex]\(2n - 1\)[/tex].
Here, the nth term is 189.
To find [tex]\(n\)[/tex], we set up the equation:
[tex]\[
2n - 1 = 189
\][/tex]
3. Solve for [tex]\(n\)[/tex]:
Add 1 to both sides of the equation:
[tex]\[
2n - 1 + 1 = 189 + 1
\][/tex]
[tex]\[
2n = 190
\][/tex]
Now, divide by 2:
[tex]\[
n = \frac{190}{2}
\][/tex]
[tex]\[
n = 95
\][/tex]
Thus, there are 95 terms in the series.
4. Use the Sum Formula:
The sum of the first [tex]\(n\)[/tex] odd numbers can be found using the formula:
[tex]\[
S = n^2
\][/tex]
Substitute [tex]\(n = 95\)[/tex] into the formula:
[tex]\[
S = 95^2
\][/tex]
5. Calculate the Sum:
Compute [tex]\(95^2\)[/tex]:
[tex]\[
95 \times 95 = 9025
\][/tex]
Therefore, the sum of the series [tex]\( 1 + 3 + 5 + \cdots + 189 \)[/tex] is [tex]\( 9025 \)[/tex].
In summary:
- Number of terms ([tex]\(n\)[/tex]): 95
- Sum of the series ([tex]\(S\)[/tex]): 9025
