Express the sum 3 + 5 + 7+ ... + (2k-1) using E-notation.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Express the sum 3 + 5 + 7 + ... + (2k-1) using -notation.

Transcribed Image Text:**Express the sum $3 + 5 + 7 + \ldots + (2k-1)$ using $\Sigma$-notation.**
### Explanation
The given sum is an arithmetic series of odd numbers starting from 3. Each successive term increases by 2.
1. **Identify the Pattern:**
- The sequence starts at 3 and follows: 3, 5, 7, ..., (2k-1).
- The general term for the odd numbers can be expressed as $a_n = 2n + 1$, but here we start at 3, so each term is really shifted by 2, becoming $2n + 3 - 2 = 2n + 1$ for $n = 1, 2, \ldots, k$.
2. **Express in $\Sigma$-notation:**
- The $\Sigma$-notation is used to succinctly express the sum of terms that follow a specific pattern.
- The expression becomes $\sum_{n=1}^{k} (2n+1)$.
### Conclusion
By using the $\Sigma$-notation, the sum $3 + 5 + 7 + \ldots + (2k-1)$ can be expressed as:
$$\sum_{n=1}^{k} (2n+1)$$
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