Theory:

In the previous theory, we understand that Arithmetic Progress \(A.P\) form a sequence of a,a+d,a+2d,a+3d,a+4d,a+5d. Here each number is called a term.
 
The first term is '\(a\)', the second term is '\(a + d\)' which is obtained by adding the difference \((d)\), and the third term is '\(a+2d\)'.
 
The terms of an \(A.P.\) can be written several ways. Now let's see the few ways which are:
General \(n^t\)\(^h\) term:
When  \(n ∈ N\), \( n = 1, 2, 3, 4, ……\),
 
\(t_1 = a = a + (1 - 1) d\)
 
\(t_2 = a + d = a + (2 - 1) d\)
 
\(t_3 = a + 2d = a + (3 - 1) d\)
 
\(t_4 = a + 3d = a + (4 - 1) d\)
 
Here '\(t\)' refers to terms, and '\(n\)' denotes the number of terms.
 
In general, the \(n^t\)\(^h\) term denoted by \(t_n\) can be written as \(t_n =  a + (n - 1) d\).
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 l=a+(n1)n=l1d+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 first term \(t_1 = a\) and the second term \(t_2 = a + d\).
 
Difference between \(t_1\) and \(t_2\) is \(t_2 - t_1 = (a + d) - a = d\).
 
Similarly, \(t_2 = a + d\) and \(t_3 = a + 2d\).
 
Therefore, \(t_3 - t_2 = a + 2d - a + d = d\).
 
So, in general \(d = t_2 - t_1 = t_3 - t_2 = t_4 - t_3 = t_5 - t_4\).
 
Thusd=tntn1 where \( n = 1, 2, 3, ……\)
The common difference of an \(A.P.\) can be positive, negative or zero.
Example:
1. Consider an \(A.P.\) \(10, 13, 16, 19, 22\)...
 
\(d = t_2 - t_1 = t_3 - t_2 = t_4 - t_3 = t_5 - t_4\).
 
\(d = 13-10 = 16-13 = 19-16 = 22-19 =3\).
 
Here the common difference is \(3\).
 
 
2. Take an \(A.P.\) \(-7, -10, -13, -16,\)..
 
\(d = t_2 - t_1 = t_3 - t_2 = t_4 - t_3 = t_5 - t_4\).
 
\(d = -10-(-7) = -13-(-10) = -16-(-13) = -3\).
 
Here the common difference is \(-3\).
 
 
3. If an \(A.P\) is \(-7, -7, -7, -7, -7\)..
 
\(d = t_2 - t_1 = t_3 - t_2 = t_4 - t_3 = t_5 - t_4\).
 
\(d = -7-(-7) = -7-(-7) = -7-(-7) = -7-(-7) =0\).
 
Here the common difference is \(0\).
 
 
An Arithmetic progression having a common difference of zero is called a constant arithmetic progression. For example, here the \(A.P\) \(-7, -7, -7, -7, -7\)..is called constant arithmetic progression.
Condition for three numbers to be in \(A\).\(P\).
If \(a\), \(b\), \(c\) are in \(A\).\(P\). then \(a = a\), \(b = a +d\), \(c = a +2d\)
 
So, \(a + c\) \(= 2a + 2d = 2 (a + d) = 2b\)
 
Thus, \(2b = a + c\)
 
Similarly, if \(2b = a +c\), then \(b − a = c −b\) so \(a, b, c\) are in \(A.P.\)
 
Thus three non-zero numbers \(a, b, c\) are in \(A\).\(P\). if and only if \(2b = a + c\)
Example:
If 4\(+\) \(x\)10\(-\) \(x\), 4\(x\) \(+\) 2 are in \(A\).\(P\). then find \(x\).
 
Since the given \(A\).\(P\). series has three numbers, we can use the above-derived expression \(2b = a + c\).
 
Let us take 4+x,10x,4x+2abc
 
Now substitute the known values in the expression \(2b = a + c\).
 
2(10x)=4+x+4x+2202x=6+5x206=5x+2x14=7x147=xx=2
 
The value of \(x\) \(=\) 2
 
Important!
Key takeaways:
  • The common difference of an \(A.P.\) can be positive, negative or zero.
  • The common difference of constant \(A.P.\) is zero.
  • If '\(a\)' and '\(l\)' are the first and last terms of an \(A.P.\) then the number of terms \((n)\) is n=l1d+1