UNIT 12.1. Patterns and Algebra

This unit focuses on different methods of solving real-life problems through the following topics- sets, sequences and series, binomial theorem and determinants.

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Topic 5: Sequences and Series

Subtopic 1: Listing sequences

Many real-world processes generate lists of numbers. For instance, the balance in a bank account at the end of each month forms a list of numbers when tracked over time. Mathematicians call such lists sequences.

Often, in Mathematics work, groups of three or more numbers are encountered that are arranged in definite order or sequence, so the each number of the group may be obtained from one or more of the preceding numbers by some fixed laws. Such a group or sequence is called a series.

Of the many kinds of sequences and series in this section we shall discuss only simplest namely, arithmetic and geometric. We shall also discuss summation notation.

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Learning outcome:

After completing this topic, you will be able to:

1. Review the basic concepts in listing a sequence and a series

2. Find the nth terms in a sequence from a given formulae

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Read the notes given below.

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Sometimes we use a letter to indicate a particular term of a sequence.a1 = first term a2 = second term a3 = third term

and in general is an = nth term.

Thus, in our example a1 = 2 a2 = 4 a3 = 6 a4 = 8 a5 = 10 and an = 2n

To find the 103rd term of this sequence, we let n = 103 to get a103 = 2 (103) = 206

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Now look at the examples of finding the terms of a sequence.

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Study these examples on series

Examples 1 and 2 are finite series while example 3 is an infinite series. A finite series is associated with a finite sequence in the same way that an infinite series is associated with an infinite sequence.An infinite series is not a number but rather an expression which may or may not sum up to a number.

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PRACTICE

Refer to the notes on slides 4 & 5, and the examples on slides 6 –8 to answer the following questions

1. Write the correct term or terms on the spaces provided.

a)A collection of numbers arranged in a particular order is called _____________.

b) The indicated sum of the terms of a sequence is called the ______________ associated with the sequence.

c) When a sequence has a last term, it has a(n) ___________________ sequence.

d) The nth term of a sequence is called the _______________ term of the sequence.

e) The numbers in a sequence are called ________________ of the sequence.

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