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Arithmetic Progression (AP) is a mathematical series in which the difference between any two subsequent numbers is a fixed value.
For example, the natural number sequence 1, 2, 3, 4, 5, 6,... is an AP because the difference between two consecutive terms (say 1 and 2) is equal to one (2 -1). Even when dealing with odd and even numbers, the common difference between two consecutive words will be equal to 2.
In simpler words, an arithmetic progression is a collection of integers where each term is resulted by adding a fixed number to the preceding term apart from the first term.
For eg:- 4,6,8,10,12,14,16
We can notice Arithmetic Progression in our day to day lives too, for eg:- the number of days in a week, stacking chairs, etc.
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Types of Progressions
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In general, progressions can be classified into 3 types:
- Arithmetic Progression (AP): An AP is a collection of numbers where each succeeding number is found either by adding or subtracting a distinct number called a common difference.
General form: m, m + n, m + 2n, ...
- Geometric Progression (GP): GP stands for Geometric Progression which is a series of numbers where each succeeding number is formed by multiplying a distinct number called a common ratio.
General form : m, mn,mn2, ...
- Harmonic Progression (HP): A series of numbers is called a harmonic progression if the reciprocal of those numbers are in an arithmetic progression.
For Eg: A, B, C are terms assigned to AP, GP, and HP respectively, so:
General form: B2 =AC
Common Terminologies in AP
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- nth term (an)
- The sum of the first n terms (Sn)
- Common Difference (d)
For a specific series in an AP, some of the commonly used terms are a common difference (d), the first term (a), and the nth term.
Let's assume the below-mentioned sequence is in an AP:
m1, m2, m3, .... mn
- To find the common difference (d)
we use the formula
d = mn - mn-1
i.e d = m2 - m1, m3 - m2, m4 - m3, etc
In general, we can conclude AP is
m, m+d, m+2d, m+3d, ... m+(n-1)d
where "m" represents the first term of an arithmetic progression.
- Finding the nth term of an AP
Formula:
an = a + (n-1)d,
where ‘a’ is the first term, ‘d’ is the common difference, and ‘n’ denotes the number of terms.
Also, note that the sequence of an AP depends on its common difference (d)
- In case if d = positive then the terms of AP will tend towards the positive side of infinity.
- In case if d = negative then the terms of AP will tend towards the negative side of infinity.
Example: Find the nth term of the AP
1, 2, 3, 4, 5…., an, if the total number of terms is 15.
Solution: It has been given that AP: 1, 2, 3, 4, 5…., an
n=15
By the formula we know, an = a+(n-1)d
So, first-term, a = 1
Common difference, d = 2-1 = 1
Therefore, an = 1 + (15-1)1
= 1+14 = 15
Sum of N Terms in an AP Sequence
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We can easily calculate the sum of n terms for any given progression.
For eg: In any AP, the sum of the first 'n' terms can be calculated if we know the first term and the total no. of terms.
Formula:
Let us assume, an AP consists of “n” terms.
S = n/2[2a + (n − 1) × d]
Example: Let’s take an example of the addition of natural numbers up to 15 numbers.
AP = 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15
We can deduce,
a = 1,
d = 2-1 = 1
and an = 15
Now, by the formula we know;
S = n/2[2a + (n − 1) × d]
= 15/2[2.1+(15-1).1]
S = 15/2[2+14] = 15/2 [16]
= 15 x 8
S = 120
Therefore, the value of the sum of the first 15 natural numbers is 120.
Sum of AP when the First and Last Terms are Given
The formula that is used to find the sum of AP when the first and last terms of the sequence are given:
S = n/2 (first term + last term)
A list of useful formulas
Some of the useful formulas to solve conceptual problems based on the series and sequence are-
| General Form of AP | a, a+d, a+2d, a+3d, . . . |
| The nth term of AP | an = a+(n – 1).d |
| Sum of n terms in AP | S = n/2[2a + (n − 1) × d] |
| Sum of all the terms present in a finite AP having the last term as ‘l’ | n/2(a + l) |
Frequently Asked Questions (FAQs)
Ques: Which of the following among the list forms an AP? If they do form an AP, write down the next two terms of the sequence : (5 marks)
(i) 4, 10, 16, 22, ...
(ii) 1, – 1, – 3, – 5,...
(iii) – 2,2,–2,2,–2,...
(iv) 1,1, 1,2,2,2,3,3,3,...
Solution :
(i) 4, 10, 16, 22, . . .
Here, we have a2–a1 = 10–4
= 6
a3–a2 = 16-10
= 6
a4–a3 = 22 – 16
= 6
This means that ak + 1 – ak is the same every time.
Hence, the given sequence of numbers forms an AP with the common difference d equal to 6.
So, the next two terms are: 22 + 6 = 28
and 28 + 6 = 34.
(ii) a2–a1 = – 1–1 = – 2
a3 – a2 = – 3 – ( –1 ) = – 3 + 1 = – 2
a4– a3 = – 5 – ( –3 ) = – 5 + 3 = – 2
This means that ak + 1 – ak is the same every time
Therefore, the given sequence of numbers forms an AP with the common difference d equal to – 2.
The next two terms are:
– 5+(–2) = –7
and – 7+(– 2 ) = – 9.
(iii) a2–a1 = 2–(–2) = 2 + 2 = 4
a3–a2 = –2–2 = – 4
As a2–a1 is not equal to a3 – a2, the given sequence of numbers doesn’t form an AP.
(iv) a2–a1 = 1–1 = 0
a3–a2 = 1–1 = 0
a4–a3 = 2–1 = 1
Here in this sequence, a2 – a1 = a3 – a2 is not equal to a4 – a3
Therefore, the given sequence of numbers doesn’t form an AP.
Ques: Find the value of n. If a=10, d=5, an=95. (2 marks)
Solution:
Given, a = 10,
d = 5,
an = 95
From the formula of general term, we have:
an = a+(n − 1).d
95 = 10 + (n − 1) × 5
(n − 1) × 5 = 95 – 10 = 85
(n − 1) = 85/ 5
(n − 1) = 17
n = 17 + 1
n = 18
Ques; Determine the AP sequence wherein the 3rd term is equal to 5 and the 7th term equates to 9. (3 marks)
Solution :
We have
a3 = a+(3 – 1).d
= a+2d
= 5 - (1)
and
a7 = a + (7–1).d
= a + 6d
= 9 - (2)
Solving both the linear equations 1 and 2, we get
a = 3, d = 1
Therefore, the required AP sequence is 3, 4, 5, 6, 7,...
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