\( \DeclareMathOperator{cosec}{cosec} \)

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Here are some specific activities, investigations or visual aids we have picked out. Click anywhere in the grey area to access the resource.

- Matchstick Patterns Create a formula to describe the nth term of a sequence by examining the structure of the diagrams.
- Arithmetic Sequences An exercise on linear sequences including finding an expression for the nth term and the sum of n terms.
- Arithmetic Sequences Video A reminder of how to find the next term, the nth term and the sum of terms of an arithmetic or linear sequence.
- Venn Diagram of Sequences Find the formula for the nth term of sequences that belong in the given sets.
- Sequences Table Challenge Complete the table showing the terms of the sequences and the formulas for the nth terms.
- Parts of Sequences Find the formula that describes the part of the sequence that can be seen

Here are some exam-style questions on this statement:

- "
*(a) Find the \(n\)th term of the sequence 7, 13, 19, 25,...*" ... more - "
*The first three and last terms of an arithmetic sequence are \(7,13,19,...,1357\)*" ... more - "
*An arithmetic sequence is given by 6, 13, 20, …*" ... more - "
*In an arithmetic sequence, the fifth term is 44 and the ninth term is 80.*" ... more - "
*A celebrity football match is planned to take place in a large stadium.*" ... more - "
*An arithmetic sequence has first term 99 and common difference \(-5.5\).*" ... more - "
*A Grecian amphitheatre was built in the form of a horseshoe and has 22 rows.*" ... more - "
*Consider an arithmetic sequence where \(u_{10}=S_{10}=15\). Find the value of the first term, \(u_1\) and the value of the common difference, \(d\).*" ... more

Here is an Advanced Starter on this statement:

Click on a topic below for suggested lesson Starters, resources and activities from Transum.

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:

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

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