To express the series $\frac{1}{20} + \frac{1}{25} + \frac{1}{30} + \frac{1}{35} + \frac{1}{40} + \frac{1}{45}$ in sigma notation, we need to find a general pattern for the terms.

Observations:

• The denominators are increasing by 5: 20, 25, 30, 35, 40, 45.
• The general form for the denominators is $20 + 5(n - 1)$, where $n$ starts from 1.
  • For $n = 1$, the denominator is 20.
  • For $n = 2$, the denominator is 25.
  • And so on.

Thus, the general term for the series can be written as:

$\frac{1}{20 + 5(n-1)}$

Sigma Notation:

The sum of this series can be written in sigma notation as:

$\sum_{n=1}^{6} \frac{1}{20 + 5(n-1)}$

Here, the lower bound $n = 1$ and the upper bound $n = 6$ corresponds to the six terms in the series.

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Would you like a breakdown of this process or any other clarification?

Here are 5 related questions:

1. How do you convert a sequence of terms into sigma notation?
2. How would you adjust the sigma notation if the series started from a different number, say $\frac{1}{10}$?
3. What are other examples of series that can be written in sigma notation?
4. How do you handle more complex denominators in sigma notation?
5. Can you find the sum of the series from sigma notation?

Tip: When identifying a pattern in the sequence, focus on how the terms change. Look for a common difference or ratio for arithmetic or geometric sequences.