Sequences and Series

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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.