Get a Clear Understanding of Arithmetic and Geometric Progression
Grasping the concepts of sequence, series, arithmetic progression and geometric progression (as well as their means) will become easy after you read through this article.
The common and most frequent question which every student has is
‘What is the difference between Sequence and Series?’
In this article, we broadly focus on sequence and series to ease problem solving. Let’s first start with sequence, and then we’ll move on to series.
- What is a Sequence?
A group of numbers which can be arranged in a definite order following a certain rule is called a sequence.
The various numbers occurring in a sequence are called its terms.
a. 2, 3, 5, 7, 11, 13, 17,… is a sequence.
The terms used in the above sequence are prime numbers. Prime numbers can be easily determined by the definition, but there is no formula to determine the same.
This means that we can determine the nth prime number with the method of selecting each successive number and check whether it is prime or not.
b. 0, 1, 0, 1, 0, 1 is a sequence of alternate 0s and 1s.
A sequence which contains a finite number of terms is called a finite sequence, and a sequence which contains an infinite number of terms is called an infinite sequence.
From the above examples,
(i) is an infinite sequence and
(ii) is a finite sequence.
- What is a Series?
Let a1, a2, a3,…, an,… be the given sequence. Then, the expression
a1 + a2 + a3 + … + an is called a series associated with the given sequence, and it is abbreviated as.
Basically, when the terms in a sequence are connected to each other by a positive or negative sign, the sequence becomes a series.
A series is finite or infinite if the given sequence is finite or infinite.
a. 2 + 3 + 6 + 8 +… is an infinite series.
b. 1 + 2 + 3 + 4 is a finite series.
c. – 1 – 2 – 3 – 4 – 5… is an infinite series.
- What is Progression?
A progression is a sequence of numbers in which each term is related to its predecessor and successor by a uniform law.
Progressions are also arranged in a definite order following a certain rule, but it has a specific formula to calculate its nth term.
24, 21, 18,…
an/tn = 27 – 3n
By using the above formula, you can find out the nth term of the given progression.
Now, we will see two important types of progression—arithmetic progression and geometric progression.
1. Arithmetic Progression
An arithmetic progression (AP) is a sequence of numbers in which each term can be obtained by adding a constant number to its preceding term. The constant number is called the common difference of an AP.
i. 5, 7, 9, 11, 13,…
ii. 18, 16, 14, 12,…
The standard form of an AP is given by
a, a + d, a + 2d,…
The general term or the nth term of an AP is an/tn = a + (n – 1)d
a = first term, d = common difference, an/tn is the nth term and n ϵ N.
For an AP,
if d > 0, then the AP is increasing
if d < 0, then the AP is decreasing
if d = 0, all the terms of an AP are the same
In the above examples, (i) is increasing as the value of d = 2 > 0 and (ii) is decreasing as the value of d = –2 < 0.
The sum of the first n terms of an AP is denoted by Sn and it is given by the formula
If the first and last terms of an AP are given, then the sum of the n terms is given by.
Properties of an AP
If a constant is added or subtracted from each term of an AP, then the resulting sequence is also an AP.
If each term of an AP is multiplied or divided by a constant, then the resulting sequence is also an AP.
2. Geometric Progression
A geometric progression (GP) is a sequence in which each term can be obtained by multiplying or dividing its preceding term by a fixed quantity. The fixed quantity is called the common ratio.
a. 8, 24, 72, 216,…
The standard form of a GP is given by
a, ar, ar2, ar3,…
The general term or the nth term of a GP is an/tn = arn-1.
a = first term, r = common ratio, an/tn is the nth term and n ϵ N.
Sum of the n terms of a GP is given by
Sn = , when |r|< 1
Sn = , when |r|> 1
Sn = na, when r = 1
Properties of a GP
If a, b and c are in GP,
i.e. b2 = ac
And if b2 = ac, then a, b and c are in GP.
If each term of a GP is multiplied or divided by a constant, then the resulting sequence is also a GP.
Now, we will see the concepts of arithmetic mean and geometric mean.
If for two numbers x and y, a number z is inserted between them so that x, z and y are in AP, then the number z= is called the arithmetic mean, whereas the geometric mean of the numbers x and y is the number.
Relation between AM and GM is Arithmetic mean is always greater than or equal to geometric mean.
i.e. AM ≥ GM
Arithmetic Progression and Geometric Progression are important topics in both CBSE Class 10 and ICSE Class 10 syllabus. Knowing them well will help you in your board exams as well as in your further studies. TopperLearning provides live classes, weekly tests and study tools to help students practise application-based problems related to progression. Also get expert advice and your doubts cleared in 24 hours by subject experts or call our counsellor on 1800-212-7858.
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