Notes — ShuleDirect

Simple number patterns

Sequences

A Sequence is the arrangement of numbers or is a list of numbers following a clear pattern such that one number and the next are separated by comma (,).

Example: a₁, a₂, a₃, a₄……………………..

NB: Each number found in a Series or Sequence is called a term.

Example 10

Find the next three terms in the following sequences.

5, 8, 11, 14, 17,………………………………
3, 7, 6, 10, 9, …………………………………
1, 2, 4, 7, ………………………………………
2, 9, 20, 35, …………………………………

Solution:

(a)You can see that each term is less to the next by 3.

So next three terms are (17+3),(17+3+3) and 17+3+3x3)

Which are 20, 23, and 26

Alternately add 4 and subs tract 1. The sequence then extends to 13, 12, 16

We see that the difference is increasing by 1 each time. So the next three terms are 11, 16 and 22.

The differences are increased by 4 each time, so the next three terms are 54, 77 and 104.

Example 11

Write down the first three terms in the sequences where the n^th term is given by the formulae.

Example 12

The k^th term of a series is k²+ 4

Find the sum of the first four terms in the series

Solution:

k=1, k²+4=1²+4=5

k=2, k²+4=2²+4=8

k=3, k²+4=3²+4=13

k = 4, k²+4 = 4²+4 = 20

So the series is 5+8+13+20 and its sum is 46

Example 13

Find the n^th term of the following sequences:

Exercise 2

1. Write down the next three terms in the following sequences.

2. Find the first three terms in the sequence:

5n+2
1-3k
n²+n+1
2ⁿ

Arithmetic sequences

When the series or sequence is such that between two consecutive terms there is a difference which is fixed, then the series or sequence is called an arithmetic progression (A.P)

The fixed difference (number) between two consecutive terms is called the common difference (d)

Example 14

In the sequence 4, 7, 19, 13, 16 there is a common difference which is

7-4=10-7=13-10=16-13=3

So the common difference (d)=3.

Note that in arithmetic progression (A.P) the difference between two successive terms is always the same.

Sometimes numbers may be decreasing instead of increasing, the arithmetic sequence or series while terms decrease have a negative number as a common difference.

Example 15

The common difference of the sequence 6, 4, 2, 0, -2, …………………… is?

In general if A₁, A₂, A₃, A₄, ……………………… A_n are the terms of the arithmetic sequence , then the common difference is ;

Example 16

For each of the following sequences, find the common difference and write the next two terms.

Solution:

Exercise 3

1. Find the common difference for each of the following sequence:

11, 14, 17, 20, …………………………………
2, 4, 6, 8, 10, ……………………………………
0.1, 0.11, 0.111, 0.1111 , …… … … … … …
y, y+3, y+6, y+9, y+12, … …… … … … ……

2. State whetherthe following sequence are arithmetic or not:

2, 5, 8, 11, 14, …………… ……………… ……
1, 3, 4, 6, 7, 9, 10, ………………………………
y, y + x, y+2x, y+3x, … ………… ……

3. The temperatureat a mid day is 3⁰c, and it falls by 2⁰c each hour. Find the temperature at the end of the next four hours.

Geometric Progression (G.P).

When the series or Sequence is such that between two consecutive terms there is a ration which is fixed, then the series or sequence is called a geometric progression (G.P)

The fixed ratio(number) between two successive terms is called the common ratio (r)

Example 17

In 2, 4, 8, 16, 32, … … …… … … …….

There is a common ration which is

Note that like in arithmetic progression (A.P), in geometric progression (G.P) the common ratio does not change.

Also the terms may be decreasing instead of increasing, the geometric sequence or series whose terms decrease have a positive common ratio which is less than 1 for the progression with positive terms.

Example 18

For each of the following sequence find the common ratio.

Example 19

For the following geometric sequences, find the common ratio and write down the next two terms:

The next term is found by multiplying the term considered to be the last term by the common ratio.

Exercise 4

1. Which of the following sequences are geometric

1, 2, 4, 8, 16, ……………………………………
2, 6, 18, 54, 162, …………………………………
1, -1,1,-1,1, ………………………………………
x², 2x³, 4x4, 8x³…………………………………
1, 2, 4, 7, 10, ………………………………………
0.1, 0.2, 0.3, 0.4, 0.5, ……………………………
3, 6, 9, 12,15, ……………………………………….

2. Find thecommon difference for each of the following geometric progressions (G.P)

3. Find thenext term of the sequence 2, 10, 50, 500,………………….

4. The populationof a town is decreasing so that every year the population declines by a quarter. If the population is originally 100,000. What will it be after 5 years?

Geometric sequence

Determining a term in a sequence

Arithmetic progression (A.P)

If A₁, A₂, A₃, …………………A_nare the terms of an arithmetic sequence, then there is a common difference d which is given by

d = A₂ – A₁ = A₃ – A₂ = A_n – A_n – 1

But . A_{3 =}A₁ + 2d which means

A₄=[ A₁+2d]+d

= A₁ + 3d

Putting into consideration this pattern, it is true that

A₅= A₁ + 4d

A₆= A₁ + 5d

A_n = A₁ + (n-1)d

Where A_n is the n^th term

The n^th term of the sequence with first term A₁and common difference d is given by

Example 20

Find theformula for the n^th term of the sequence 8 , 9.5, 11, 12.5, 14, 15.5,……

Note that the n^th term gives every term in the sequence,

For example when n=3, you have A₃=1.5x3+6.5=11

So A₃=11 where 11 is given in the sequence above having the third position.

Therefore A_n shows the position of the term in sequence and of A₁+(n-1)d gives the value of the term for any positive integer.

Example 21

The 5^th term of an arithmetic sequence is 11, and the 8^th term is 26. Find the first five terms.

Example 22

The 8^th term of an arithmetic sequence is 9 greater than the 5^th term, and the 10^th term is 10 times the 2^nd term. Find

The common difference (d)
20^th term.

Example 23

Find the formula for the n^th term of each of of the following geometric sequence.

2, 6, 18, 54 , ………………………………
4,-2, 1, -0.5, 0.25 …………………………

Geometric progression (G.P)

Sum of an A.P

When the terms are separated by addition (+) sign, there we have what we call a series.

Example: 2+4+6+8+……………………………

Is a series with the first term (A₁) 2 and common difference (d) 2

It is possible to establish a formula for the sum of the first n terms of the arithmetic progression.

Let S_n denote the sum of the first n terms of the arithmetic series.

Consider the sum of the first 5, terms of arithmetic progression (AP) whose first term is 1 and whose common difference (d) is 1.

So S₅= A₁+A₂+A₃+A₄+A₅

S₅=1+2+3+4+5 ………………………….. (1)

The first case is the sum of five terms which are increasing from 1 up to 5 while the second case shows the same sum but the terms are decreasing from 5 to 1.

If you add (1) and (2) together, you find that

S₅+S₅=(1+5) + (2+4) +(3+3) + (4+2) + (5+1)

25_s= 6+6+6+6+6

25₅=30

Dividing by 2 each side gives

Also we can write

Example 24

Find the sum of the first 20 terms of the series.

Example 25

Find the sum of the series 4+7+10+13+…………….+304

Solution:

To use the formula for summation of n terms, you must know how many terms are there, i.e finding the value of n;

Now

A₁=4, d=3 and A_n = 304 n=?

A_n =A₁+ (n-1)d

304 = 4+ (n-1)x3

304 = 4+3n-3

304=3n+1

304=3n

Example 26

How many terms of the series 1+3+5+7+………………. are needed to make the sum of 169?

Exercise 5

1. Find thesum of the first 20 terms of the series

2+5+8+11+……………………
19+16+13+10+7+……………

2. Find thenumber of terms and the sum of the series:

1+3+5+7+ ……………………………………
40+37+34+31+…………………+-257

3. The sumof the first 10 terms of an arithmetic progression (A.P) is 40, and the sum of the next 10 terms is 80. Find the sum of the first five terms of the series.

4. One dayFrola spends 40 minutes of her home work. The length of time she spends increase by 4 minutes each day. Find the total length of time she spends after eight days.

Sum of a G.P

Geometric series are the series that can be written as

G₁+G₂+G₃+ ………………………..G_n

Example: 2+4+8+16+ ………………………..+G_n

Or 1+3+9+27+81+…………………….

Suppose we want to find the sum of 1+3+9+9+27+81+…………………

S₅=!+3+9+27+81……………………….(1)

If we multiply s_n by the common ratio(r), we have.