Number and Algebra

Furthermore

Official Guidance, clarification and syllabus links:

Spreadsheets, GDCs and graphing software may be used to generate and display sequences in several ways. If technology is used in examinations, students will be expected to identify the first term and the common difference.

Examples of applications include simple interest over a number of years.

Where a model is not perfectly arithmetic in real life students will need to approximate common differences.

Formula Booklet:

The nth term of an arithmetic sequence	\( u_n = u_1 + (n-1)d \)
The sum of n terms of an arithmetic sequence	\( S_n = \frac{n}{2}(2u_1+(n-1)d); \\ S_n = \frac{n}{2}(u_1 + u_n) \)

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.

If you use a TI-Nspire GDC there are instructions here for generating a sequence on the calculator.