# Sequences and Types Arithmetic progression

**SECOND TERM**

**WEEK 1 DATE(S)……………………………………….**

**SUBJECT: FURTHER MATHEMATICS**

**CLASS: SS1**

**TOPIC: SEQUENCES**

**CONTENT**:

- Sequences and Types
**.** - Arithmetic progression
- Geometric progression

**Sub – Topic: SEQUENCE AND TYPES**

A sequence is a succession of terms generated according to some rules or laws. Examples are: (a) 1, 3, 5, 7 (b) 2, 4, 6, 8, 10, … (c) 2, 4, 8, 16, 32, … (d) ^{2}, ^{3}, ^{4}, ^{5}, …

Every member of each set is called a term. The three dots after each set show that the set of the numbers or terms continues definitely. The terms of a sequence are sometimes called elements of the sequence. When the number of terms in a sequence is limited, the sequence is said to be finite. On the other hand, if the number of elements in a sequence is unlimited or otherwise, the sequence is said to be infinite.

Finite sequences may be specified by listing the elements of the sequence and this list may be complete or incomplete. They can also be specified by a functional relation.

An infinite sequence can also be specified by an incomplete list of its terms or by a functional relation. Sequence (a) above is an example of a finite sequence. It has four terms. The sequences (b), (c) and (d) above are examples of infinite sequences. We may write the terms of a sequence as _{1}, _{2}, _{3}, _{4}, …

The general terms or the th term can be designated _{k}. The th term is _{n} by this designation.

**Example: **

- Given that
_{k2}, (. write the first four terms of the sequence.

**Solution: **

_{k21223} . Hence, the first four terms of the sequence are

A sequence of numbers can be generated in any fashion as long the pattern is consistent. When we can identify the pattern of the sequence, our immediate task will be to find a formula for the general term of the sequence which is usually designated as T_{n} and it is usually a function of

Two important types of sequences are the **arithmetic sequence** and **geometric sequence. **The pattern of generating the arithmetic sequence is by adding or subtracting a constant number to a preceding term to get the terms while the geometric sequence is generated by multiplying or dividing a constant number with a preceding term to get the terms.

Other popular types of sequences include: **Triangular numbers** and **Fibonacci sequence. **A triangular number is an integer that can be depicted by a triangular array of dots.

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**1 3 6 10**

The sequence 1, 3, 6, 10, 15, 21,… constitute triangular numbers and it is neither arithmetic nor geometric. If T_{n} is nth term of the sequence, it can be shown that: **T _{n} = .** Note that the sum of two consecutive terms of triangular number is a square, i.e.. E.g. , , i.e.

The relation above is called a **recurrence relation. **Consider the sequence 1,1,2,3,5,8,13,… , generated from recurrence relation. This sequence is called **Fibonacci sequence. **It was discovered by **Fibonacci**, an Italian mathematician, in 1202 A.D.

**Example:**

Find the second and fourth terms of the sequence whose general term is given by:

- b) c)
^{n}d)^{n-2}e)

**Solution:**

- T
_{n}: T_{2}and T_{4} - T
_{n}T_{2}and T_{4} - T
_{n}^{n}: T_{2}^{2}and T_{4}^{4} - T
_{n}^{n-2}T_{2}^{2-2}and T_{n}^{4-2} - T
_{n}: T_{2}and T_{n}

A sequence of numbers can be generated in any fashion as long as the pattern by which the terms are generated is consistent. Two importance ways by which a sequence can be generated are:

- By adding or subtracting a constant number to a preceding term to get a term. A sequence generated in this way, is called linear sequence or an arithmetical progression.
- By multiplying or dividing a preceding term by a constant number to get a term. A sequences generated in this way is called an exponential sequence or geometrical progression.

**CLASS ACTIVITY**

- Define sequence
- Explain with examples (a) finite and (b) infinite sequence.
- Mention two ways a sequence can be generated and write out their names.
- A sequence of numbers
_{1},_{2},_{3},_{4n}satisfies the relation_{n}=_{n-1}+ for all positive integers if_{1}=1, - Write down the expression for
_{1, 2}and_{4} - Obtain an expression for
_{n}in its simplest form. - Find an expression value of
_{n }for large values of n.

**Sub – Topic: LINEAR SEQUENCE OR ARITHMETIC SEQUENCE**

This is a sequence in which each term differs from the preceding term by a constant amount called the common difference.

If the first term is and the common difference is then the sequence takes the form: Thus, if _{1}, _{2}, _{3}, _{4n} are the n terms of the sequence then: _{1 }; _{2}; _{3}; _{4}** . . .** ;** _{n}**. Linear sequence or arithmetic sequence is sometimes called an arithmetic progression(AP).

**Examples:**

- Find the common difference in each of the following arithmetic sequence:
- 5, 8, 11, 14, 17, …
- 83, 77, 71, 65, …

**Solutions:**

Let the common difference be then:

- The eleventh term of an A.P is 25 and its first term is find its common difference. (WAEC).

**Solution:**

Let the th term be _{n} and the first term Let be common difference. Then:_{n}

**Arithmetic Mean**: Suppose is a number between two numbers such that are in arithmetical progression. The number is called the arithmetic mean of Thus, if are in arithmetical progression, then the common difference is given as:

. Also,

**Example:**

- Find the arithmetic mean of 6 and 24.

**Solution:**

Let

- Find given that are terms of an arithmetic sequence.

**Solution:**

_{n}5.

**Class activity: **

- Find the 6
^{th}term of an arithmetic sequence whose first term is 3 and whose common difference is 5. - The first term of an A.P. is equal to twice the common difference d. Find, in terms of d, the 5
^{th}term of the A.P.

**Sub – Topic: EXPONENTIAL OR GEOMETRIC SEQUENCE(G.P):**

This is a sequence in which the ratio of each term (except the first) to the preceding term is a constant. This constant is called the common ratio. If the first term is and the common ratio is then the sequence has the form _{2}, _{3}, …

Thus, _{1}^{0}; _{2}^{1}; _{3}^{2}; _{4}^{3}. . . ** _{n}^{n-1}**. Hence, the th term is

_{n}

^{n-1}

**Examples:**

- Find the common ratio in each of the following

**Solutions: **

let

- The common ratio of a G.P is 2. If the fifth term is greater than the first term by 45, find the 5
^{th}term (WAEC).

**Solution:**

Let: _{5}^{4}.

_{5}^{44.}

Also, _{55}^{4}

Class activity: Find the 10^{th} term of the sequence 96, 48, 24, …

**Geometric progression mean:** Let be consecutive terms of an exponential sequence and the common ratio, then by definition

Equating (i) and (ii)

^{2}

We call the geometric mean of

**Examples:**

- Insert two geometric mean between 6 and 162.

**Solution:**

Let and be the two geometric means between 6 and 162. Then consecutive terms of a geometric progression.

_{1}and_{4}^{3}

Dividing (i) and (ii) we have: ^{3}_{2}

_{3}^{2}

**CLASS ACTIVITY:**

- I f are in Geometric Progression, find the product of
- The first and third terms of a G.P. are 5 and 80 respectively. What is the 4
^{th}term?

**PRACTICE EXERCISE**

- Which term of the sequence 128, 64, 32 …, is ?
- The fifth, ninth and sixteenth terms of a linear sequence(A.P) are consecutive terms of an exponential sequence(G.P).
- Find the common difference of the linear sequence in terms of the first term.
- Show that the twenty – first, thirty – seventh and sixty – fifth terms of the linear sequence are consecutive terms of an exponential sequence whose common ratio in the common ratio is .
- Find the common ratio in the G.P:
- The first and last terms of an A.P are 2 and 125 respectively. If the fifth term is 14, find the number of terms in the A.P. (WAEC).
- Find the nth term of the sequence 2, 12, 36, 80…

**ASSIGNMENT**

- A sequence
_{r }is defined as_{r }= and . - Find the values of a and b
- Using (i) obtain the general formula for
_{r }in terms of r only. - Consider the Fibonacci sequence 1,1,2,3,5,8,13,21,34, … where
_{n+2 }=_{n }+_{n+1 }.Find the next two terms - The 10th, 4th and 1st terms of an A.P. are the three consecutive terms of a G.P. Find the common ratio of the G.P.
- If the sum of the 8th and 9th terms of a linear sequence is 72 and the 4th term is – 6 , find the common difference.
- Insert three arithmetic mean between – 3 and 5

**KEY WORDS**

**SEQUENCE****COMMON (RATIO/DIFFERENCE)****PROGRESSION (ARITHMETIC OR LINEAR/GEOMETRIC OR EXPONENTIAL)****MEAN(ARITHMETIC/GEOMETRIC)****TERM(S) {n**^{th}term/general term}

**PRE-READING ASSIGNMENT:**

Study sequence and series using Further Mathematics Project 1 by M. R. Tuttuh-Adegun, et al.