To write the series $49 + 54 + 59 + \ldots$ for 14 terms using summation notation, we first recognize the pattern of the sequence.

Step 1: Identify the arithmetic sequence components

1. First term ($a$): $49$
2. Common difference ($d$): $54 - 49 = 5$
3. General term ($a_n$): The $n$-th term of an arithmetic sequence is given by: $a_n = a + (n - 1)d$ Substituting $a = 49$ and $d = 5$: $a_n = 49 + (n - 1) \cdot 5 = 49 + 5n - 5 = 5n + 44$

Step 2: Write the summation notation

The series for 14 terms is the sum of the first 14 terms of the arithmetic sequence: $\sum_{n=1}^{14} a_n = \sum_{n=1}^{14} (5n + 44)$

Thus, the series in summation notation is: $\sum_{n=1}^{14} (5n + 44)$

Let me know if you'd like a step-by-step explanation or a deeper dive into summation properties!

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Expanding the Topic

Here are 5 related questions:

1. How do you calculate the sum of an arithmetic series?
2. Can you write a series with a common difference of 7 for 20 terms in summation notation?
3. What is the difference between summation notation and expanded form?
4. How can you generalize an arithmetic sequence if given any two terms?
5. How would you find the 100th term of this series?

Tip

Always double-check the common difference when working with arithmetic sequences—it ensures your formula is correct!