![]() ![]() But if the first differences are NOT the same, and instead, the second differences are the same, then the sequence is known as a quadratic sequence.Įxample: The sequence 1, 2, 4, 7, 11. We have already seen that if the differences (referred to as first differences) between every two successive terms are the same, then it is called an arithmetic sequence (which is also known as a linear sequence). This fixed number is called a common difference. The succeeding terms are obtained by adding a fixed number, that is, $3. So, the amount in her piggy bank follows the pattern of $30, $33, $36, and so on. She increased the amount on her each successive birthday by $3. Įxample: Mushi put $30 in her piggy bank when she was 7 years old. The terms of the arithmetic sequence are of the form a, a d, a 2d. ![]() Arithmetic SequenceĪn arithmetic sequence is a sequence of numbers in which each successive term is a sum of its preceding term and a fixed number. ![]() We will discuss these sequences in detail. and this sequence does not belong to any of the following sequences. is a sequence in which the numbers can be written as 1 3 1, 2 3 1, 3 3 1, 4 3 1. Apart from these, there can be sequences that follow some other pattern. At the end of the first year you will have a total of: \ With simple interest, the key assumption is that you withdraw the interest from the bank as soon as it is paid and deposit it into a separate bank account.There are a few special sequences like arithmetic sequence, geometric sequence, Fibonacci sequence, harmonic sequence, triangular number sequence, square number sequence, and cube number sequence. You are paid $15\%$ interest on your deposit at the end of each year (per annum). We refer to $£A$ as the principal balance. Simple and Compound Interest Simple Interest For example, \ so the sequence is neither arithmetic nor geometric. A series does not have to be the sum of all the terms in a sequence. The starting index is written underneath and the final index above, and the sequence to be summed is written on the right. ![]() We call the sum of the terms in a sequence a series. The Summation Operator, $\sum$, is used to denote the sum of a sequence. If the dots have nothing after them, the sequence is infinite. If the dots are followed by a final number, the sequence is finite. Note: The 'three dots' notation stands in for missing terms. is a finite sequence whose end value is $19$.Īn infinite sequence is a sequence in which the terms go on forever, for example $2, 5, 8, \dotso$. For example, $1, 3, 5, 7, 9$ is a sequence of odd numbers.Ī finite sequence is a sequence which ends. Contents Toggle Main Menu 1 Sequences 2 The Summation Operator 3 Rules of the Summation Operator 3.1 Constant Rule 3.2 Constant Multiple Rule 3.3 The Sum of Sequences Rule 3.4 Worked Examples 4 Arithmetic sequence 4.1 Worked Examples 5 Geometric Sequence 6 A Special Case of the Geometric Progression 6.1 Worked Examples 7 Arithmetic or Geometric? 7.1 Arithmetic? 7.2 Geometric? 8 Simple and Compound Interest 8.1 Simple Interest 8.2 Compound Interest 8.3 Worked Examples 9 Video Examples 10 Test Yourself 11 External Resources SequencesĪ sequence is a list of numbers which are written in a particular order. ![]()
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