THE CONCEPT OF SEQUENCE AND SERIES
A sequence is a set of numbers in which the numbers have a prescribed order. Example of sequence

1, 3, 5, 7, 9, 11, 13 …

2, 4, 8, 16, 32, 64 …

A series is a sum of all the terms of sequence. Example of series

1+3+5+7+9+11+13+…

2+4+8+16+32+64+…

TERMS OF A SEQUENCE/SERIES
These are the numbers that form a sequence or a series and they are denoted by , where ……….

= the first term

= the second term

= the third term

= the last term or n^{th}-term or general term

TYPES OF SEQUENCE/SERIES
In this level, we only observe two types of sequence/series namely:-

Arithmetic Progression (A.P)

Geometric Progression (G.P)

ARITHMETIC PROGRESSION (A.P)
Arithmetic progression (A.P): where the difference between successive terms is constant/common, and is denoted by (this is a key property of A.P)
The terms of A.P sequence/series are denoted by , where …..
Suppose that are the terms of A.P, then
GEOMETRIC PROGRESSION (G.P)
Geometric Progression (G.P): where the ratio between successive terms is constant/common (this is a key property of G.P)
The terms of G.P sequence/series are denoted by , where …..
Suppose that are the terms of G.P, then Example:#000000
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Given the series …………………. Say whether it is an AP or GP.

Solution: Case 1:
Assume the series …………………. follows A.P, then

Condition for A.P is a common difference ( :

Case 2:
Assume the series …………………. follows G.P, then

Condition for G.P is a common ratio ( :

Observing the two cases above, the series satisfy the A.P condition, hence it is an A.P. THE GENERAL TERM OF AN ARTHMETIC PROGRESSION (A.P)
The General term of A.P is given by:
PROOF:
Given that: are the terms of A.P, it follows that;

Given the series 100+92+84+……………… find the 20^{th} term.

Solution:
Check if the series follows an A.P or not
For Arithmetic progression (A.P);
In our case:

It follows that:

The series is an A.P, then 20^{th} term is obtained by using the below formula
,

THE GENERAL TERM OF A GEOMETRIC PROGRESSION (G.P)
The general term of G.P is given by: PROOF:
Suppose that, are the terms of G.P, it follows that
⤇
⤇
⤇ : : : : : : , Hence proved.

Solution:
………………….. (i)
………….. (ii)
………… (iii)
………. (iv)
Substituting (i) into (iv), we get

From (ii):

From (iii): . THE SUM OF N-TERMS OF ARTHIMETIC PROGRESSION (A.P)
Suppose that the series ……………………. is an A.P, then its sum is given by:- Or

PROOF:
Given an A.P series …………………. . Then its sum ( ) is
…………………….
……… --------------- (1)
.......... + --------------- (2)
Equation (1) and (2) mean the same thing; it is just a matter of arrangement.
Adding equation (1) and (2), simplifies to
………+

The sum of the eighth term and tenth term of an Arithmetic progression AP is 28 and the sum of the first ten terms is 35; find the first term and the common difference of this A.P

Solution:

=28
…………………….…….. (i)

…………………………….. (ii)
Solving simultaneously equation (i) and (ii) for and , we get;
From equation (i)

…………………… (iii)
Substituting equation (iii) into (ii), we get;

From equation (iii):

The first term ( ) = and common difference ( ) = THE ARITHMETIC MEAN
The arithmetic mean of A.P sequence is given by PROOF:
Suppose that; are sequence terms of A.P and is an arithmetic mean, it follows that:
…………………………….. (Key property of A.P)

Solution:
Given two terms of A.P; and
Arithmetic Mean (A.M);

THE SUM OF THE FIRST N-TERMS OF GEOMETRIC PROGRESSION (G.P)
Suppose that the series is a G.P, then its sum is given by:-
or PROOF:
Given an G.P series . Then its sum ( ) is

. --------------- (1)
Multiply by the common ratio ® on both sides of equation (1), we get;
. ------------- (2)
Now, consider the two cases: CASE 1:
Subtract equation (1) from (2), we get

, Hence proved. CASE 2:
Subtract equation (2) from (1), we get

, Hence proved.
NB: The two formulas are nothing but one, they are set two basically for those who get problems when dealing with negative numbers. Example:#000000
solid;margin:15px;width:95%;">[td]The sum of the first two terms of a geometric progression is 10 and the sum of the first four terms is 40. Given that all terms of the progression are positive, show that.

The common ratio is

The sum of the first n terms is 5( ).

[/td]

Solution:

From the formula: …………. (1)

When,
=10 …………. (i)
When,
= 40

………….. (ii)
Substituting equation (i) into (ii) we get,

………….. (iii)
Given that all terms of the progression are positive, it follows that;
Hence shown.

Substituting equation (iii) into (i) we get,

……… (iv)
Substituting (iii) and (iv) into (1) we get,

Hence shown THE GEOMETRIC MEAN (G.M)
The Geometric mean of a three terms sequence of G.P is given by
PROOF:
Suppose that; are sequence terms of G.P and is a geometric mean, it follows that:
……………………….. (Key property of G.P)

The arithmetic mean and geometric mean of two numbers m and n 17 and 15 respectively. Find two numbers.

Solution:
Arithmetic Mean (A.M)

…………….. (i)
Geometric Mean (G.M)

………………… (ii)
Solving simultaneously equation (i) and (ii) we get,

………………… (iii)
Substitute equation (iii) into (ii), we get

This is a quadratic equation in , where

Or
Or

When ,

When , Or COMPOUND INTEREST
In a simple interest, the principle is taken constant each year, but under a compound interest the amount obtained at the end of a specific year is taken as a principal to its next year.

…….. Amount of a compound interest

…….. The compound interest

……… n^{th} year compound interest

Where,
= is the amount collected at the end of n-years.
= Principal

= Duration of interest rate (always a year, then )
=number of times that the money can be collected from where invested. I = the compound interest.
PROOF OF THE FORMULA:
From a simple interest:

………………… (Amount at the end of first year)
Now
Where

………………. (Amount at the end of second year)
Now P =
Where

……………… (Amount at the end of third year)
Observing the sequence ……………., it follows that; …………….. (Amount at the end of n-years). Hence proved.
………………… (The compound Interest)
The formula , is valid if and only if the principal is compounded annually. NB: if the amount is collected more than once in a year, then below formula is valid.

On the birth of a child. What sum of money should a family invest at 9% per annum, compounded semi-annually, to provide the child with Tsh 1,000,000/= on his sixteenth birth day?

Solution:

……………….. (1)
Apply log on both sides of equation (1), we get

REVIEW QUESTIONS:

Determine the sum of the first eight terms of the series 1 – ½ + ½ - ⅛ + . . .

The sum of a number of consecutive terms of an arithmetic progression is –19.5, the first term is 16.5 and the common difference is –3. Find the number of terms of the progression.

In a geometric progression, the sum of the second and the third terms is 6 and the sum of the third and fourth terms is –12. Find the first term and the common ratio of the progression.

The arithmetic mean and geometric of two numbers a and b are 17 and 15 respectively. Find the value of a and b.

Prove that the series is an arithmetic progression whose sum to n terms is

Find three numbers in geometric progression such that their sum is 39 and their product is 729.

The nth term of a certain sequence is Find the sum of the first five terms of the corresponding series.

John borrowed £1200 from Zachary with agreement of 5% compounded quarterly, what amount of money Zachary demand from John after 4 years.

Find the amount A to which the principal P amounts at R% interest compounded annually for n years is given by the formula An = P(1 + 0.01R)^{n}. If P = 126, R = 4 and n = 8

The third, the fourth and eighth terms of an arithmetic progression forms the first three consecutive terms of a geometric progression. If the sum of the first ten terms of the A.P is 85. Calculate:

(i) The first terms of both A.P and G.P
(ii) The common ratio
(iii) The sum of the first four terms of the G.P