Arithmetic Progression | Revision Notes

Arithmetic Progression | Revision Notes

Arithmetic Progression

Important Terms, Definitions and Results


  • Some numbers arranged in a definite order, according to a definite rule, are said to form a sequence.
  • A sequence is called an arithmetic progression (AP), if the difference of any of its terms and the preceding term is always the same.
    i.e., an – an – 1 = constant.
  • The constant number is called the common difference of the A.P.
  • If a is the first term and d the common difference of an AP, then the general form of the AP is a, a + d, a + 2d, ...
  • Let a be the first term and d be the common difference of an AP, then, its nth term or general is given by an = a + (n – 1) d
  • If l is the last term of the AP, then nth term from the end is the nth term of an AP, whose first term is l and common difference is – d.
    nth term from the end = Last term + (n – 1) (– d)
    nth term from the end = l – (n – 1) d
  • If a, b, c, are in AP, then
    • (a + k), (b + k), (c + k) are in AP.
    • (a – k), (b – k), (c – k) are in AP.
    • ak, bk, ck, are in AP.
    • a/k, b/k, c/k are in AP(k ≠ 0).
  • Remember the following while working with consecutive terms in an AP.
    • Three consecutive terms in an AP.
      a – d, a, a + d    
      First term = a – d, common difference = d
      Their sum = a – d + a + a + d = 3a
    • Four consecutive terms in an AP.
      a – 3d, a – d, a + d, a + 3d
      First term = a – 3d, common difference = 2d
      Their sum = a – 3d + a – d + a + d + a+ 3d = 4a
    • Five consecutive terms in an AP.
      a – 2d, a – d, a, a + d, a + 2d
      First term = a – 2d, common difference = d
      Their sum = a – 2d + a – d + a + a + d + a + 2d = 5a
  • The sum Sn up to n terms of an AP whose first term is a and common difference d is given by
    S _ { n } = \frac { n } { 2 } [ 2 a + ( n - 1 ) d ]
  • If the first term and the last term of an AP are a and l, then
    S _ { n } = \frac { n } { 2 } \left( a + l \right) = \frac { n } { 2 } ( \text { first term } + \text { last term } )
  • an = Sn – Sn–1