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 + ……
…………………………………
n3 – (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….