Arithmetic Progression:
The sequence of numbers in which each term differ by a constant number from the previous term through out the sequence is known as arithmetic progression.
Let \(a\) and \(d\) be real numbers.
Then the numbers of the form \(a, a+d, a+2d, a+3d, a+4d, a+5d,...\) is said to form Arithmetic progression. And it is denoted by \(A.P\).
The number ‘\(a\)’ is called the first term and ‘\(d\)’ is called the common difference.
Here each number in the sequence is called a term.
The first term is '\(a\)', the second term is '\(a + d\)' which is obtained by adding the common difference \((d)\), the third term is '\(a+2d\)' and so on.
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 \(n^{th}\) term is denoted by \(t_n\) and 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
Common difference:
To find the common difference of an \(A.P\), 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 \(d = t_2 - t_1 = t_3 - t_2 = t_4 - t_3 = t_5 - t_4\).
Thus, where \( n = 1, 2, 3, ……\)
The common difference of an \(A.P.\) can be positive, negative or zero.
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 \(2b = a + c\).
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