Grade 11 Mathematics Note

Sequence and Series and Mathematical Induction.

The idea of a sequence originates in the process of counting in a very natural way. A sequence is an ordered list of numbers. The sum of the terms of a sequence is called a series.

Types of sequences

a. Arithmetic sequences

In an Arithmetic Sequence the difference between one term and the next is a constant. In other words, we just add the same value each time ... infinitely.

Example: 1, 4, 7, 10, 13, 16, 19, 22, 25, ...

This sequence has a difference of 3 between each number. 

We can write an Arithmetic Sequence as a rule:

Xn = a + d(n-1)

(We use "n-1" because d is not used in the 1st term).

Example: Write the Rule, and calculate the 4th term for

3, 8, 13, 18, 23, 28, 33, 38, ...

This sequence has a difference of 5 between each number.

The values of a and d are:

a = 3 (the first term)

d = 5 (the "common difference")

The Rule can be calculated:

Xn = a + d(n-1)

= 3 + 5(n-1)

= 3 + 5n - 5

= 5n - 2

So, the 4th term is:

X4 = 5*4 - 2 = 18

Arithmetic sequence: a1, (a1 +d), (a1 +2d),……..,(an-d), an

Arithmetic series: a1+ (a1 +d)+ (a1 +2d)+……..+(an-d), an.

 

A sequence such as 1, 5, 9, 13, 17 or 12, 7, 2, –3, –8, –13, –18 which has a constant difference between terms. The first term is a1, the common difference is d, and the number of terms is n.

 

Explicit formula: an = a1 + (n – 1)d.

Example:

3, 7, 11, 15, 19 has a1 = 3, d = 4, and n = 5.

The explicit formula is an = 3 + (n – 1)4

= 4n – 1

Geometric sequence and series

A geometric sequence is a sequence of numbers that follows a pattern were the next term is found by multiplying by a constant called the common ratio, r.

an = an-1.r or an = a1 . rn-1

Example

Write the first five terms of a geometric sequence in which a1=2 and r=3.

We use the first given formula:

a1=2a1=2

a2=2⋅3=6a2=2⋅3=6

a3=6⋅3=18a3=6⋅3=18

a4=18⋅3=54a4=18⋅3=54

a5=54⋅3=162a5=54⋅3=162

Just as with arithmetic series it is possible to find the sum of a geometric series.

The sum of the first n terms of geometric series is given by

${{\rm{s}}^{\rm{n}}} = \frac{{{{\rm{a}}_1}\left( {1 - {{\rm{r}}^{\rm{n}}}} \right)}}{{1 - {\rm{r}}}}$ When r is smaller than 1

${{\rm{s}}^{\rm{n}}} = \frac{{{{\rm{a}}_1}\left( {{{\rm{r}}^{\rm{n}}} - 1} \right)}}{{{\rm{r}} - 1}}$ When 1 is smaller than r

And also ${{\rm{s}}^{\rm{n}}} = \frac{{{{\rm{a}}_1} - {{\rm{a}}_{\rm{n}}}.{\rm{r}}}}{{1 - {\rm{r}}}}$

 

Example:

The given series is, 1 + 4 + 9 + 16 + ….

i.e. 12 + 22 + 32 + 42 + ….

So the nth term, tn= n2.

Let sn be the sum of the 1st’n’ terms, then

Sn = ∑tn∑tn⁡ = ∑n2∑n2

So, Sn = 12 + 22 + 32 + 42 + …. + n2 …(1)

Let, r3 – (r – 1)3 = 3r2 – 3r + 1

Put r = 1, 2, 3, 4, ….n. We have,

13 – 03 = 3.12 + 3.1 + 1

23 – 13 = 3.23 – 3.2 + 1

33 – 23 = 3.32 – 3.3 + ……

…………………………………

n– (n – 1)3 = 3.n2 – 3.n + 1

Adding, we have.

n3 – 03 = 3(12 + 22 + 32 + …. + n2) – 3(1 + 2 + 3 + …. + n) + n.1

or, n3 = 3Sn – 3n(n+1)2 + n

or, 3Sn = n3 + 23n(n+1)2 - n

= n2 (2n2 + 3n + 3 – 2) = n2 (2n2 + 3n + 1)

= n2 (2n2 + 2n + n + 1)

Or, 3Sn = n2 {2n(n + 1) + (n + 1)}

So, sn = n(n+1)(2n+1)6.

 

Means

If G is the geometric mean between a and b, then

= 1G2−a2+1G2−b2=1ab−a2+1ab−b2

= 1a(b−a) – 1b(b−a) = b−aab(b−a) = 1ab = 1G2 = R.H.S.

Infinite series:

The sum of infinite terms that follow a rule

When we have an infinite sequence of values:

                                               $\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{{16}}, \ldots .$

Which follow a rule (in this case each term is half the previous one),

And we add them all up:

                             $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} +  \ldots  +  \ldots  = s$

We get an infinite series.

Example: Sum to infinity the following series:

1:

1 – 12 + 14 – 18 + …

Soln:

The given series is, 1 – 12 + 14 – 18 + …

So, a = 1, r = -12, S∘ = ?

We have, S∘S∘ = a1−4 = 11+12=23.

 

2:

16 – 8 + 4 + …

Soln:

The given series is, 16 – 8 + 4 + …

So, a = 16, r = −12, S∘S∘ = ?

We have, S∘S∘ = a1−r = 161+12= 16∗23 = 323.

 

Mathematical induction

The principle of mathematical induction states that if p(n) be the statement and if

1. P(1) is true

2. P(k+1) is true whenever P(K) is true

Then P(n) is true for all nεN….

 


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