Hello friends, earlier we have studied about Proportional and non-proportional linear relationships. Now today we all are going to learn about the next topic that is terms in arithmetic sequences. It is one of the important areas of study which not just play an important role in eighth grade but also in our higher grades. Before proceeding further let's talk about the basic concept behind Sequence. In simplest mathematics manner it is a systematic list of objects. In mathematics the sequence list can be numbers, algebraic expressions, fractions etc. If the objects in the list are numbers then it is called as numeric sequence. Moving forward we will discuss numeric sequence topic in detail. Hence the sequence with number we mean the numeric sequence. Now each of the number used to form a sequence is called a term. In a sequence there can be finite number of terms or it can have infinite number of terms as well. Finite number of sequences can be stated as finite sequence and a sequence with infinite number of terms can be stated as infinite sequence. If we try to add a series of sequences then we get a series. So the total of the terms used in the sequences is called a series.
Let's take an example to understand the basic concept behind numeric sequences. A series : 1,3,5,7,9 is a finite number sequence with 5 terms. The another expression that is 3 + 6 + 9 + 12 + 15 + 18 is also a finite numeric sequence with six terms.
Similarly we can easily identify the infinite sequences like 3,1,6,4,5,4,...... etc.
Now moving further we are going to see the different types of sequences. There are various kinds of sequences in mathematics, but in eighth standard we all are going to discuss about three types of sequences.
For example : 2,4,6,8,10 ….... , 5,7,9,......................, etc.
In the above two sequences, each term is different from the previous term by a constant number and this statement is always true for any of the two consecutive terms. This kind of difference is known as the common difference. In simple words any of the term used in an arithmetic sequence is obtained by adding a constant number to the previously used term.
The most common form of an arithmetic sequence with the first term is “b” and the common difference d is given by
b, b + d, b + 2d,…………………………….b + (n – 1) d,………..
It is noticeable that first term is “b”, second term in the sequence is b + d and the third term in the above mentioned sequence is b + 2d etc.... So in this manner we proceed further and the nth term will be
b + (n – 1)d. In short we can say that an arithmetic sequence can be defined by the first term and the common difference.
,100,200,400,800……………………….., 7,21,63,189……………………
In both of the above examples each term used in the sequence maintains a definite or we can say a constant ratio with the previous term. This can be stated in a more defined manner as if we take any term in the sequence and try to divide it by the previously formulated term we get the same number. This type of numbers in a sequence is called that common ratio of the geometric sequence.
In normal way we can understand it in a manner that any term of a geometric sequence is obtained by multiplying a constant number to the previous term
The common form of an arithmetic sequence with first term “a” and common ratio that is “r” is given by
a, ar,ar2,ar3,ar4………..ar(n-1)………….
Here it is also noticeable that the first term used is “a”, the second term in use are ar, and the third term in the sequence is ar2. So the nth term used in the sequence will be ar(n-1)
In short we can say that a geometric sequence is defined by the first term and the common ratio
Let's take an example to understand it better : ½,1/3, ¼, 1/5, 1/6 …...... 1/5, 1/10, 1/15.... etc.
The most common form of the harmonic sequence can be expressed from the general form of the arithmetic sequence used by taking the reciprocal of the terms.
Let's take some example to illustrate the above mentioned terms and methodology:
Example 1: Here the problem is that the first term of an arithmetic sequence is 4 and the fifth term of the sequence is 20. Find the sequence
Solution: Let the first term b = 4
By using above mentioned sequence or general form we can find the fifth term from the formula for nth term b+ (n-1) d
Now the fifth term is b+(4-1)d= 20
b+3d= 20
4+4d=20
4d = 20-4=16
d= 16/4=4
So b=4 and d=4 the sequence is
4,8,12,16,20.........
Example 2: The next question is that the first term used in a geometric sequence is 6 and the fourth term used is 48. We need to find out the basic sequence form :
Solution : Let the first term is “a” = 6
Now the fourth term that can be calculated from the formula for nth term ar(n-1)
The fourth term which is to be used is ar(4-1) = ar3= 48
ar3= 48
6r3=48
r3= 48/6=8 so we can get r=2
So a=6 and r=2 the sequence we get after formulating this problem is :
6,12,18,24,..........
The other topic which we are going to understand consist of following terms that are Variables, expressions, equations and inequalities. Now take individual topic at a time :
Inequality: An inequality tells that two values are not equal. For example a ≠ b shows that a is not equal to b. Slope formula plays an important role in graphing linear inequalities. So we need to know what slope formula means. Slope of a line describes the steepness, incline or grade of the straight line.
Equation: An equation is basically explained as an assertion that two algebraic expressions are equal. Let's take an example: 3a + 1 = 4
Here we can get
Let's take an example to understand the basic concept behind numeric sequences. A series : 1,3,5,7,9 is a finite number sequence with 5 terms. The another expression that is 3 + 6 + 9 + 12 + 15 + 18 is also a finite numeric sequence with six terms.
Similarly we can easily identify the infinite sequences like 3,1,6,4,5,4,...... etc.
Now moving further we are going to see the different types of sequences. There are various kinds of sequences in mathematics, but in eighth standard we all are going to discuss about three types of sequences.
-
Arithmetic Sequence :
For example : 2,4,6,8,10 ….... , 5,7,9,......................, etc.
In the above two sequences, each term is different from the previous term by a constant number and this statement is always true for any of the two consecutive terms. This kind of difference is known as the common difference. In simple words any of the term used in an arithmetic sequence is obtained by adding a constant number to the previously used term.
The most common form of an arithmetic sequence with the first term is “b” and the common difference d is given by
b, b + d, b + 2d,…………………………….b + (n – 1) d,………..
It is noticeable that first term is “b”, second term in the sequence is b + d and the third term in the above mentioned sequence is b + 2d etc.... So in this manner we proceed further and the nth term will be
b + (n – 1)d. In short we can say that an arithmetic sequence can be defined by the first term and the common difference.
-
Geometric Sequence
,100,200,400,800……………………….., 7,21,63,189……………………
In both of the above examples each term used in the sequence maintains a definite or we can say a constant ratio with the previous term. This can be stated in a more defined manner as if we take any term in the sequence and try to divide it by the previously formulated term we get the same number. This type of numbers in a sequence is called that common ratio of the geometric sequence.
In normal way we can understand it in a manner that any term of a geometric sequence is obtained by multiplying a constant number to the previous term
The common form of an arithmetic sequence with first term “a” and common ratio that is “r” is given by
a, ar,ar2,ar3,ar4………..ar(n-1)………….
Here it is also noticeable that the first term used is “a”, the second term in use are ar, and the third term in the sequence is ar2. So the nth term used in the sequence will be ar(n-1)
In short we can say that a geometric sequence is defined by the first term and the common ratio
-
Harmonic sequence :
Let's take an example to understand it better : ½,1/3, ¼, 1/5, 1/6 …...... 1/5, 1/10, 1/15.... etc.
The most common form of the harmonic sequence can be expressed from the general form of the arithmetic sequence used by taking the reciprocal of the terms.
Let's take some example to illustrate the above mentioned terms and methodology:
Example 1: Here the problem is that the first term of an arithmetic sequence is 4 and the fifth term of the sequence is 20. Find the sequence
Solution: Let the first term b = 4
By using above mentioned sequence or general form we can find the fifth term from the formula for nth term b+ (n-1) d
Now the fifth term is b+(4-1)d= 20
b+3d= 20
4+4d=20
4d = 20-4=16
d= 16/4=4
So b=4 and d=4 the sequence is
4,8,12,16,20.........
Example 2: The next question is that the first term used in a geometric sequence is 6 and the fourth term used is 48. We need to find out the basic sequence form :
Solution : Let the first term is “a” = 6
Now the fourth term that can be calculated from the formula for nth term ar(n-1)
The fourth term which is to be used is ar(4-1) = ar3= 48
ar3= 48
6r3=48
r3= 48/6=8 so we can get r=2
So a=6 and r=2 the sequence we get after formulating this problem is :
6,12,18,24,..........
The other topic which we are going to understand consist of following terms that are Variables, expressions, equations and inequalities. Now take individual topic at a time :
Inequality: An inequality tells that two values are not equal. For example a ≠ b shows that a is not equal to b. Slope formula plays an important role in graphing linear inequalities. So we need to know what slope formula means. Slope of a line describes the steepness, incline or grade of the straight line.
Let's take an example to understand the linear inequalities: How to solve a compound inequality and graph the solutions?
Example: -6 < 2x - 4 < 12
Example: -6 < 2x - 4 < 12
-6 < 2x - 4 < 12
add 4 to all 3 parts
-2 < 2x <16
divide 2 from all 3 parts
-1 < x < 8
add 4 to all 3 parts
-2 < 2x <16
divide 2 from all 3 parts
-1 < x < 8
To graph the following equation, you put an open circle or we can say mark it by a dot on the point (-1,0) and then you put an open circle on the point (8,0).Then draw a line between the 2.
We need to be careful while solving inequalities, as they are harder to solve than equations and require more attention. You can multiply an equation by a positive or negative number and get an equivalent equation. But while solving Inequalities remember this: when multiplying it by a negative number, you need to change sign of the inequality.
A linear inequality describes an area of the coordinate plane that has a boundary line. In simple way in linear inequalities everything is on one side of a line on a graph.
In mathematics a linear inequality is an inequality which involves a linear function. For solving inequalities we need to learn the symbols of inequalities like the symbol < means less than and the symbol > means greater than and the symbol รข‰¦ or ≤ less than or equal to etc.Equation: An equation is basically explained as an assertion that two algebraic expressions are equal. Let's take an example: 3a + 1 = 4
Here we can get
a = 1 is the solution. This is an example of a linear equation. An example of a quadratic equation is a2 - a - 2 = 0. This equation has the two solutions
a = -1 or a = 2.
Algebra and Variable relationship: If we talk in simple words then we will see that algebra is simply the art of replacing variables in place of numbers. In solving algebraic problems simplifying algebraic terms is important. Simplifying here refers to breaking the large expressions into smaller ones so that it becomes easy to solve.
So this is all about in our today's class. In next class we will continue with this topic. As this topic is very lengthy considering the syllabus of eighth standard.
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