1.1 Given the Arithmetic sequence: $$ 10+15+20 $$ 1.1 .1 How many terms are there in the series? 1.12 . Write down the series in sigma notation 1.13 Calculate the number of the series.

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Let's break down the problem step by step.

### Step 1: Identify the known conditions
We have an arithmetic sequence given by the terms $$10, 15, 20$$.

### Step 1.1: How many terms are there in the series?
In an arithmetic sequence, the difference between consecutive terms is constant.

- The first term $$a = 10$$
- The second term $$a + d = 15$$ where $$d$$ is the common difference.
- The third term $$a + 2d = 20$$

To find the common difference $$d$$:
$$
d = 15 - 10 = 5
$$

Now, we can see that the sequence continues as $$10, 15, 20, 25, \ldots$$.

To find the number of terms in the series, we need to determine the last term. However, since the problem does not specify a last term, we will assume it continues indefinitely.

If we consider the terms given, we have:
- $$10$$ (1st term)
- $$15$$ (2nd term)
- $$20$$ (3rd term)

Thus, there are **3 terms** in the series provided.

### Step 1.2: Write down the series in sigma notation
The general term of an arithmetic sequence can be expressed as:
$$
a_n = a + (n-1)d
$$
where $$a$$ is the first term, $$d$$ is the common difference, and $$n$$ is the term number.

For our sequence:
- $$a = 10$$
- $$d = 5$$

The $$n$$-th term can be expressed as:
$$
a_n = 10 + (n-1) \cdot 5
$$

The series can be written in sigma notation as:
$$
\sum_{n=1}^{3} (10 + (n-1) \cdot 5)
$$

### Step 1.3: Calculate the number of the series
To calculate the sum of the series, we can use the formula for the sum of the first $$n$$ terms of an arithmetic series:
$$
S_n = \frac{n}{2} \cdot (a + l)
$$
where:
- $$n$$ is the number of terms,
- $$a$$ is the first term,
- $$l$$ is the last term.

In our case:
- $$n = 3$$
- $$a = 10$$
- $$l = 20$$

Now, substituting these values into the formula:
$$
S_3 = \frac{3}{2} \cdot (10 + 20)
$$

Calculating this:
$$
S_3 = \frac{3}{2} \cdot 30 = \frac{90}{2} = 45
$$

### Final Answers
1.1.1: There are **3 terms** in the series.
1.1.2: The series in sigma notation is $$\sum_{n=1}^{3} (10 + (n-1) \cdot 5)$$.
1.1.3: The sum of the series is **45**.
There are 3 terms in the series. The series in sigma notation is $$\sum_{n=1}^{3} (10 + (n-1) \cdot 5)$$. The sum of the series is 45.

1.1 Given the Arithmetic sequence: \( 10+15+20 \) 1.1 .1 How many terms are there in the series? 1.12 . Write down the series in sigma notation 1.13 Calculate the number of the series.

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