Express the sum 3 + 5 + 7+ ... + (2k-1) using E-notation.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Express the sum 3 + 5 + 7 + ... + (2k-1) using -notation.

**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:

$$\sum_{n=1}^{k} (2n+1)$$
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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