Arithmetic Progression:
The sequence of numbers in which each term differs by the common difference from the previous term throughout the sequence is known as an arithmetic progression.
Important!
Let \(a\) and \(d\) be the first term and real numbers. Then the numbers of the form \(a\), \(a + d\), \(a + 2d\), \(a + 3d\), \(a + 4d\), \(a + 5d\), ... are said to form an Arithmetic Progression, and it is denoted by \(A.P\).
Arithmetic Progression in real-life:
In a theatre, the arrangement of seats forms an arithmetic Progression. For instance, the first row might have \(12\) seats, the second row \(14\) seats, and the third row \(16\) seats.
General \(n^{th}\) term:
In general, the \(n^{th}\) term denoted by \(t_n\) can be written as \(a_n = a + (n -1) d\).
In a finite \(A.P\) whose first term is \(a\) and last term \(l\), then the number of terms in the \(A.P\) is given by \(n = \left (\frac{l-a}{d}\right ) + 1\).
Common difference:
To find the common difference of an \(A.P\) generally, we should subtract the first term from the second term, the second from the third and so on.
The common difference of an \(A.P\) can be positive, negative or zero.
Example:
Consider an \(A.P\) \(10\), \(13\), \(16\), \(19\), \(22\), .....
\(d = t_2 - t_1 = t_3 - t_2 = t_4 - t_3 = t_5 - t_4\)
\(d = 13 - 10 = 16 - 13 = 19 - 16 = 22 - 19 = 3\).
An arithmetic progression having a common difference of zero is called a constant arithmetic progression.
Condition for three numbers to be in \(A.P\):
If \(a\), \(b\), \(c\) are in \(A.P\) if and only if \(2b = a + c\).
Example:
Check whether the three numbers \(2\), \(4\) and \(6\) are in \(A.P\)
Solution:
Consider \(a = 2\), \(b = 4\) and \(c = 6\)
The condition for three numbers to be in \(A.P\) is \( 2b = a + c\)
\(2(4) = 2+ 6\)
\(8 = 8\)
Therefore, three numbers satisfy the given condition. So these numbers are in \(A.P\).
Sum of the consecutive terms:
If the sum of three consecutive terms of an \(A.P\) is given, then they can be taken as \(a - d\), \(a\), and \(a + d\), and the common difference is \(d\).
Sum of four consecutive terms:
If the sum of four consecutive terms of an \(A.P\) is given then, they can be taken as \(a - 3d\), \(a -d \), \(a + d\) and \(a + 3d\). Here, the common difference is \(2d\).
Arithmetic series
An arithmetic series is a series whose terms are in Arithmetic progression.
The sum of first '\(n\)' terms of a Arithmetic progression is given by \(S_n = \frac{n}{2}[2a + (n - 1)d]\)
If the first term \(a\) and the last term \(l\) are given then, the sum of '\(n\)' terms is given by \(S_n = \frac{n}{2}[a + l]\).
Suppose we have the series \(a_1, a_2, a_3,... a_n,... \), then the number of terms in the series is neither negative nor zero. The number of terms in the series is always positive.
Example:
Find the sum of the series \(7, 13, 19, 25, 31, ...\) upto \(22\) terms.
The given series is \(7, 13, 19, 25, 31, ...\)
Here \(a = 7\), \(d = 13 - 7 = 6\), \(n = 22\).
\(S_n = \frac{n}{2}[2a + (n - 1)d]\)
\(S_{22} = \frac{22}{2}[2\times 7 + (22 -1 ) 6]\)
\(S_{22} = 11[14 + (21 \times 6)]\)
\(S_{22} = 11 \times 140 \)
\(S_{22} = 1540\).
Therefore, the sum of \(22\) terms is \(1540\).