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Arithmetic Progression (A.P.)
Meaning of Arithmetic Progression (A.P.)
Finite and Infinite Arithmetic Progression
Positive and Negative Common Difference
Properties of Arithmetic Progression
Graphical Representation of the Arithmetic Progression
Representation on Number Line
Representation on the Cartesian Plane
Arithmetic Series
Formula of nth term of Arithmetic Series
Sum of an Arithmetic Series
Arithmetic Mean
Arithmetic Progression is the sequence of numbers such that the difference between the two successive terms is always constant. And that difference is called the Common Difference. It is also known as Arithmetic Sequence.
The first term of the sequence is called the Initial Term which is denoted as ‘a’.
The difference by which the consecutive numbers of the sequence increases or decreases is called the Common Difference, which is denoted by‘d’.
The general term that is, the nth term of the arithmetic progression with the initial term a and the common difference d is as
a_{n} = a + (n-1) d
Example
Consider the right figure-
Here the sequence is given as
1, 3, 5, 7, 9, 11, 13, 15
Given a = 1 and d = 2
Let’s check the value of the 5^{th} term using the nth term formula.
a_{5} = 1 + (5 -1)2
= 1 + 8
= 9
It is given in the right figure that the 5^{th} term is 9.
The arithmetic progression with a limited number of terms is called Finite Sequence. It has a last term.
2, 9, 16, 23,…, 86
This is a finite sequence with a = 2 and d = 7
The arithmetic progression with unlimited number of terms is called Infinite Sequence. It does not have a last term.
1, 3, 5, 7, …
This is an infinite sequence with a = 1 and d = 2
It depends upon the value of the common difference that the arithmetic progression is increasing or decreasing.
If d > 0 and satisfies that a_{n-1} < a_{n} , then
a_{1} < a_{2} < a_{3}…
11, 22, 33, 44, 55, …
This is an increasing sequence with a = 11 and d = 11.
If d < 0 and satisfies that a_{n-1} > a_{n} , then
a_{1} > a_{2} > a_{3}…
20, 10, 0, -10, -20, …
This is a decreasing sequence with a = 20 and d = -10.
If the terms of the arithmetic progression are increased or decreased with the same number then the resultant sequence will also be an arithmetic progression.
3, 6, 9,… is an AP with a = 3 and d = 3
If we will add 2 to every term of the sequence then the new sequence will be
5, 8, 11, … with a = 5 and d = 3
Hence, this is also an arithmetic progression.
If the terms of the arithmetic progression are multiplied or divide with the same number (non-zero) then the resultant sequence will also be an arithmetic progression.
1, 2, 3, 4,… is an AP with a = 1 and d = 1
If we multiply the every term of the sequence with 5 then the new sequence will be
5, 10, 15, 20, … with a = 5 and d = 5
Hence, this is also an arithmetic sequence.
The Arithmetic Progression can be shown through the graphs also. It could be on number line or on the Cartesian plane.
On the number line, the AP can be shown as follows:
Here in the above number line, it is showing the arithmetic progression
1, 2, 3, 4, 5, 6, …
with a = 1 and d = +1
AP = 3, 5, 7, 9, 11
a = 3 and d = +2
Here we need the Cartesian coordinates to represent it on the Cartesian plane.
a_{n }is the nth term, so the corresponding Cartesian coordinates will be in the form of (n, a_{n} ).
For the right arithmetic progression the coordinates will be as follows:
When we add the terms of the arithmetic progression then that sum is the arithmetic series. Like, the AP, the difference between every consecutive term is always constant.
The arithmetic series is in the form of
{a + (a + d) + (a + 2d) + (a + 3d) + .........}
where a is the first term of the series and d is the difference of it which is known as the common difference of the given series.
If a is the first term, d is the difference and n is the total number of the terms, then the formula for nth term is given by
(1, 3), (2, 5), (3, 7), (4, 9), (5, 11)
a_{n} = a + (n - 1) d
What is the 10^{th} term of the series 2 + 4 + 6 + 8 +…?
Solution:
In the given series a = 2 and d = 2
So we will put the values in the formula of nth term of the series.
Here n = 10 as we have to find the 10^{th} term.
a_{10} = 2 + (10 - 1) 2
= 2 + 18
= 20
So the 10^{th} term of the series will be 20
If a is the first term and d is the common difference of the series with n number of terms, then the sum of the series will be
Find the sum of the given arithmetic progression
2 + 5 + 8 + 11 + 14
Given
a = 2 (first term)
d = 3 (common difference)
n = 5 (as the total number of terms of the series is 5)
Now we will put the values in the formula
If we know the last term ‘l’ of the series instead of the common difference ‘d’, then the sum of the series would be
Let’s take the same example as above
Now we will find the sum of the series with a= 2 and l = 14
Hence the sum remains the same if we calculate with the formula.
Arithmetic Mean is basically the average of two numbers. If we have two numbers n and m, then we can include a number L in between these numbers so that the three numbers will form an arithmetic sequence like n, L, m.
In that case the number L is the arithmetic mean of the numbers n and m.
According to the property of Arithmetic progression, we can say that-
L – n = m – L that is, the common difference of the given AP.
This is generally used to find the missing number of the sequence between the two given numbers.
What will be the 6^{th} number of the sequence if the 5^{th} term is 12 and the 7^{th} term is 24?
As the two numbers are given so the 6^{th} number will be the Arithmetic mean of the two given numbers.
Hence the 6^{th} term will be 18.
We can insert two or more numbers between the two given numbers to form an Arithmetic Progression.
Let A_{1} , A_{2} , A_{3} , …, An be n numbers between a and b such that a, A_{1} , A_{2} , A_{3} , …,An , b is an A.P.
Here, b is the (n + 2) th term, that is, b = a + [(n + 2) – 1] d = a + (n + 1) d.
This gives
Thus, n numbers between a and b are as follows:
Insert 5 numbers between 3 and 21 such that the resulting sequence is an A.P.
Let A_{1}, A_{2}, A_{3}, A_{4} and A_{5} be the five numbers between 3 and 21 such that
3, A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, 21
the sequence is in A.P.
Here, a = 3, b = 21, n = 7.
Therefore,
21 = 3 + (7 –1) d
18 = 6d
d = 3
Thus A 1 = a + d = 3 + 3 = 6;
A 2 = a + 2d = 3 + 2 × 3 = 9;
A 3 = a + 3d = 3 + 3 × 3 = 12;
A 4 = a + 4d = 3 + 4 × 3 = 15;
A5 = a + 5d = 3 + 5 × 3 = 18;
Hence, five numbers between 3 and 21 are 6, 9, 12, 15 and 18
More Readings
https://en.wikipedia.org/wiki/Arithmetic_progression
https://www.khanacademy.org/math/in-in-grade-10-ncert/in-in-chapter-5-arithmetic-progressions/in-in-sum-of-first-n-terms-of-an-ap/v/formula-for-arithmetic-series
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