ARITHMETIC PROGRESSION-TIPS

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IMPORTANT TIPS

• Observe the following sets of numbers carefully:
  
  
  
  
  1. 1, 2, 3, 4, 5, ...
  2. 2, 4, 6, 8, 10, ...
  3. 1, 4, 9, 16, 25, ...
  4. 1, 11, 111, 1111, 11111, ...
  5. 1, 1/2, 1/3, 1/4, 1/5, ...
  
  
  
  
  Here, in each set, we find numbers arranged in some order and there is an obvious (definite) rule by which we can obtain the next number and as many subsequent numbers as we wish to find.
  
  Such sets are called SEQUENCES ( PROGRESSION ) and each number of the set is called a TERM of the sequence.



• A sequence refers to an ordered set of numbers in which each number (term) can be obtained by a definite rule.



• The rule by which a sequence is formed may be written as a formula for n^th term of the series but all sequences need not have a formula.



• The n^th term of 1, 2, 3, 4, 5, ... is n.



• The n^th term of 1, 4, 9, 16, 25, ... is n².



• The n^th term of 3, 4, 7, 12, 19, ... is (n - 1)² + 3.



• 2, 3, 5, 7, 11, 13, ... is the progression of prime numbers.



• The n^th term of a progression is denoted by either a_n or T_n



• If a progression starts with a definite real number and if any successive term is obtained by adding a constant nonzero real number to the previous term, then the progression is called an ARITHMETIC PROGRESSION (written in short as A.P.)



• Sequences (a) and (b) in the examples given in the beginning are examples of Arithmetic progression whereas sequences (c), (d) and (e) are progressions but they are not Arithmetic progressions.



• The difference between any two consecutive terms of an A.P. is a nonzero constant. This difference is called the COMMON DIFFERENCE.



• The general form of an A.P. whose first term is a (a ∈ R) and the common difference is d (d≠ 0) is:
  
  a, a + d, a + 2d, a + 3d, ...



• The n^th term (T_n) of an A.P. is
  
  T_n = a + (n - 1)d



• The sum of the first n terms of an A.P. is given by
  
  S_n = (n/2)(a + l)
  
  where a = first term of the A.P. and l = last term of the A.P.
  
  Since l = T_n = a + (n - 1)d,
  
  S_n = n/2 [ 2a + (n - 1)d]



• T_n - T_n-1 = d



• S_n - S_n-1 = T_n



• T_m - T_n = (m - n)d [where m>n,&m,n ∈ N].