$$\DeclareMathOperator{cosec}{cosec}$$

# Sequences and Series

## Furthermore

Arithmetic sequences and series are fundamental concepts in mathematics. An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference, denoted by $$d$$.

The $$n^{th}$$ term of an arithmetic sequence can be calculated using the formula:

$$u_n = u_1 + (n - 1)d \\ \text{where } u_n \text{ is the } n^{th} \text{ term, } u_1 \text{ is the first term, and } d \text{ is the common difference.}$$

The sum of the first $$n$$ terms ($$S_n$$) of an arithmetic sequence can be found using the formula:

$$S_n = \frac{n}{2}(2u_1 + (n - 1)d) \text{ or equivalently, } S_n = \frac{n}{2}(u_1 + u_n) \\ \text{where } S_n \text{ is the sum of the first } n \text{ terms.}$$

Sigma notation ($$\Sigma$$) is a concise way to represent the sum of sequences. For an arithmetic sequence, it can be represented as:

$$S_n = \sum_{r=1}^{n} (u_1 + (r - 1)d)$$

Example 1: Consider an arithmetic sequence where the first term $$u_1 = 5$$ and the common difference $$d = 3$$. Let's find the $$10^{th}$$ term and the sum of the first 10 terms.

Using the formula for the $$n^{th}$$ term:

$$u_{10} = 5 + (10 - 1) \cdot 3 = 32$$

So, the $$10^{th}$$ term is 32.

Now, using the formula for the sum of the first $$n$$ terms:

$$S_{10} = \frac{10}{2}(2 \times 5 + (10 - 1) \cdot 3) = 185$$

Thus, the sum of the first 10 terms of the given arithmetic sequence is 185.

Example 2: For an arithmetic sequence with the first term $$u_1 = 2$$ and the common difference $$d = 7$$, we can represent the sum of the first 6 terms using sigma notation as:

$$S_6 = \sum_{r=1}^{6} (2 + (r - 1) \cdot 7)$$

This notation concisely represents the sum of the sequence: 2, 9, 16, 23, 30, and 37, which equals 117.

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