Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Discrete Math:
![**Problem:**
Find the sum of the arithmetic sequence:
\(8 + 12 + 16 + 20 + \cdots + 1104\)
**Description:**
This is an arithmetic sequence where the first term \(a = 8\) and the common difference \(d = 4\). The last term of the sequence is \(1104\).
**Steps to Solve:**
1. **Identify the number of terms:**
Use the formula for the \(n\)-th term of an arithmetic sequence:
\[
a_n = a + (n-1) \cdot d
\]
Set \(a_n = 1104\) and solve for \(n\):
\[
1104 = 8 + (n-1) \cdot 4
\]
\[
1104 = 4n + 4
\]
\[
1100 = 4n
\]
\[
n = 275
\]
2. **Calculate the sum of the sequence:**
Use the sum formula for an arithmetic sequence:
\[
S_n = \frac{n}{2} \cdot (a + a_n)
\]
Substitute the values:
\[
S_{275} = \frac{275}{2} \cdot (8 + 1104)
\]
\[
S_{275} = 137.5 \cdot 1112
\]
\[
S_{275} = 152900
\]
**Conclusion:**
The sum of the sequence is \(152900\).](https://content.bartleby.com/qna-images/question/76b39ad3-98e6-4bd8-afd7-9890d91d962f/170c872c-3f08-41ab-a6ea-68ebf508f50d/swui7dg_processed.png)
Transcribed Image Text:**Problem:**
Find the sum of the arithmetic sequence:
\(8 + 12 + 16 + 20 + \cdots + 1104\)
**Description:**
This is an arithmetic sequence where the first term \(a = 8\) and the common difference \(d = 4\). The last term of the sequence is \(1104\).
**Steps to Solve:**
1. **Identify the number of terms:**
Use the formula for the \(n\)-th term of an arithmetic sequence:
\[
a_n = a + (n-1) \cdot d
\]
Set \(a_n = 1104\) and solve for \(n\):
\[
1104 = 8 + (n-1) \cdot 4
\]
\[
1104 = 4n + 4
\]
\[
1100 = 4n
\]
\[
n = 275
\]
2. **Calculate the sum of the sequence:**
Use the sum formula for an arithmetic sequence:
\[
S_n = \frac{n}{2} \cdot (a + a_n)
\]
Substitute the values:
\[
S_{275} = \frac{275}{2} \cdot (8 + 1104)
\]
\[
S_{275} = 137.5 \cdot 1112
\]
\[
S_{275} = 152900
\]
**Conclusion:**
The sum of the sequence is \(152900\).
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