SS1 Second Term Further Mathematics Lesson Note – Series

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WEEK 2

CLASS: SS1

TOPIC: SERIES

CONTENT:

1. Definition of series.
2. Arithmetic series
3. Geometric series
4. Sum to infinity

Sub – Topic: DEFINITION OF SERIES/ARITHMETIC SERIES

Series is the summation (addition) of the terms of a sequence. For example, the series equivalent of the sequence 1,2,3, 4, is 1+2+3+4+…. If the sequence contains finite numbers of terms, then the corresponding series is finite. If the sequence contains infinite numbers of terms, then the corresponding series is infinite.

Arithmetic Series:

These are series formed from an arithmetic progression. e.g.

1 + 4 + 7 + 10 + …

In general, if S_n is the sum of n terms of an arithmetic series then

S_n = a + (a + d) + (a + 2d) + … + (l – d) +l   …………………….. (1)

Where l is the nth term, a is the first term and d is the common difference.

Rewriting the series above starting with the nth/last term, we have.

S_n = l +(l – d) + (l – 2d) +… + (a + d) + a  ………………………….(2)

Adding equation (1) and (2) we have

2S_n = (a + l) + (a + l) + … + (a + l) + (a + l) in n places

2S_n = n(a + l)

\S_n = ⁿ/₂(a + l)  

But l is the nth term i.e a + (n – 1) d

S_n=  ⁿ/₂{a +a + (n – 1)d}

\S_n = ⁿ/₂ {2a + (n – 1)d}

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