Sequences & series basic concepts

February 21, 2023January 21, 2021 by Louis Nkengakah

This summary notes or let’s say quick notes on sequences & series is destined to help students of the class of form 5 get some of the key points on what sequences & series are all about/

As earlier said, this is just a summary lesson note and we will be touhing jus the key points.

We also have some short summary notes on Logarithms and indices here on Edukamer.

What are sequences and series?

A sequence

A sequence is a set of terms that progresses or regresses in a defined order. For example 1, 2, 3 …

A series

A series is formed when the terms of a sequence are added. For example 1+3+5+7.

Arithmetic progression (AP)

Qn arithmethic progression is a series or sequence which progresses or regresses with a common or constant difference between the terms denoted by d. For example, 1, 3, 5, 7… Here, the common difference d is 3 -1 or 5- 3 = 2 = d.

The terms of an AP

If the first term of an AP is a and the common difference is d, then the terms of the AP are, a, a+d, a + 2d, a + 3d … a + (n – 1)d.

Therefore, the number of terms of the AP is given by:

T_n= a + (n – 1)d

Example

Consider the series 2, +4, +6, +8 …., find the fifth term of this series.

Solution

T_n= a + (n – 1)d

The first term a = 2, d = 4 -2 or d = 6 – 4

Þ d = 2

From T_n = a + (n – 1)d

\ T₅ = 2 + (5 – 1)2 = 2 + 4(2)

⇒ T₅ = 10

Sum of the terms of an AP

The sum of the first n term of an AP whose first term is a and the common difference is d is given by;

$\inline \LARGE S_n= n/2 [2a+(n-1)d]$

Example:

Find the sum of the first 10 terms of the sequence 3, 10, 17 …

Solution:

The first term

The common difference $\inline \LARGE d=10-3=17-10 =7$

$\inline \LARGE S_1_0= 10/2 [2(3)+(10-1)7]$

$\inline \LARGE S_1_0=5[6+63] = 345$

The arithmetic mean AM

This is the average of a set of numbers which is calculated by dividing the sum of all terms by the number of terms.

Example:

Find the AM of 3, 4, 5, 6, 7

Solution:

The number of terms is 5,

$\LARGE AM = (3+4+5+6+7)/5= 25/5 =5$

If a, b, c are three consecutive terms of an AP, then b is called the arithmethic mean of a and c.

That is;

$\inline \LARGE d=b-c=c-b$

$\inline \LARGE 2b=a+c$

$\inline \LARGE \Rightarrow b=(q+c)/2$

Geometric progression GP

A geometric progression is a series or sequence which progresses or regresses with a constant multiplier called the common ratio and is denoted by r. e.g 3, 6, 12, 24 … each term is obtaioned by multiplying the previous term by 2.

The common ratio r is therefore calculated as;

$\inline \LARGE r= 6/3= 12/6= 24/12=2$

Terms of a geometric progression

If the first term of a GP is a and the common ratio is r, then the terms of the GP are: a, ar, ar², ar³, …ar^n-1

Therefore, the nth term of the GP is given by;

$\inline \LARGE T_n=ar^{^{(n-1)}}$

Example:

Find the (th term of the series 16+8+4+2

Solution:

$\inline \LARGE a=16,\: \: \: r= 8/16 \: \: or \: \: 4/8 \: \: \: or \; \: 2/4= 1/2,\, \, \: n=5$

From $\inline \large T_n=ar^{(n-1)}$

$\large T_5=(16) (\frac{1}{2})^{(5-1)}=1$

Sum of the terms of a GP

The sum of the first n terms of a GP whose first term is a and common ratio r is given by;

$\inline \LARGE S_n= \frac{a(1- r^n )}{(1-r)} ,Where \: \: \: \: r<1$

OR

$\inline \LARGE S_n= \frac{a(r^n-1)}{(r-1)} \: \: \: ,Where \: \: \: r>1$

Example:

Find the sum of the first 8 terms of the sequence 2, 6, 18 …

Solution:

$\inline \LARGE a=2,\: \: r= 6/2 \: \: or \: \: 18/6 \Rightarrow r=3. \: \: i.e \: \: r>1$

$\inline \LARGE \therefore S_n= \frac{a(r^n-1)}{(r-1)}$

$\inline \LARGE \Rightarrow S_8=\frac{2(3^8-1)}{(3-1)}=6560$

If x, y, z are the three consecutive terms of a Geometric progression, then y is the geometric mean of y and z.

$\inline \LARGE i.e\: \: \: r=\frac{y}{x} \: \: or \: \: \frac{z}{y} \Rightarrow \frac{y}{x}=\frac{z}{y} \Rightarrow y=\sqrt{xz}$

The Geometric Mean (G.M) of a series containing n observations is the nth root of the product of the values.

Consider, if x₁, x₂ …. X_nare the observation, then the G.M is defined as:

$\LARGE G.M=\sqrt[n]{x_1 x_2...x_n}$

The sum to infinity of a GP

If the number of terms in a GP is not finite, then the GP is called infinite GP.

The sum to infinity of a GP whose first term is a and common ratio r is given by ;

$\LARGE S_\infty =\frac{a}{1-r} \: \: \: where \: \: \: |a|<1$

The sum of the terms of a sequence

The nth term of a sequence which is not necessarily an AP or GP is given by

$\LARGE T_n=S_n-S_{n-1}$

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Further reading on sequences and series

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