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Basic Concepts Of Geometric Progression for IIT JEE and other Engineering Exams

Posted on - 08-02-2017

JEE Math GP P And S

IIT JEE

Geometric Progression (G.P.)

Definition

A G.P. is a sequence whose first term is non-zero and each of whose succeeding term is r times the preceding term, where r is some fixed non - zero number, known as the common ratio of the G.P. .

Term and Sum of n Terms:

If a is the first term and r the common ratio, then G.P. can be written as a, ar, ar2, . . . the nth term, an, is given by an = arn-1. The sum Sn of the first n terms of the G.P. is .

If -1 < r < 1, then the sum of the infinite G.P. a + ar + ar2 +........=

Notes:

  • If each term of a G.P. is multiplied (divided) by a fixed non-zero constant, then the resulting sequence is also a G.P. with same common ratio as that of the given G.P.
  • If each term of a G.P. (with common ratio r) is raised to the power k, then the resulting sequence is also a G.P. with common ratio rk. .
  • If a1, a2, a3, .......and b1, b2, b3, ....... are two G.P.’s with common ratios r and r¢ respectively then the sequence a1b1 , a2b2, a3b3.....is also a G.P. with common
    ratio r r¢.
  • If we have to take three terms in a G.P., it is convenient to take them as a/r, a, ar. In general, we take in case we have to take (2k+1) terms in a G.P.
  • If we have to take four terms in a G.P., it is convenient to take them as a/r3, a/r, ar, ar3 . In general, we take , in case we have to take 2k terms in a G.P.
  • If a1, a2, . . . . ,an are in G.P. , then a1 an = a2 an-1 = a3 an-2 = . . . . .
  • If a1, a2, a3, ….… is a G.P. ( each aI > 0), then loga1, loga2, loga3 ….. is an A.P. The converse is also true.

    Geometric Means:

    • If three terms are in G.P., then the middle term is called the geometric mean (G.M.) between the two. So if a, b, c are in G.P. then b = is the geometric mean of a and c.
    • If a1, a2......an are non-zero positive numbers then their G.M (G) is given by
      G = (a1a2a3
      ......an)1/n. If G1, G2,…..Gn are n geometric means between a and b then
      a, G1, G2, …
      ., Gn, b will be a G.P. Here b = a rn + 1 .

    • &⇒
      r =
      &⇒
      , Gr = awhere Gr is the rth mean.

      Example 1

      The third term of a G.P. is 7. Find the product of first five terms.

      Solution:

      Let the terms be , a , ar, ar2
      &⇒
      a = 7.

      The product = a5 = 75.

      Example 2

      Show that is a composite number.

      Solution:

      We have = 1090 +1089 + 1088 + . . . … . + 102 + 10 +1

      = = = m ´ n , where m and n are natural numbers ( an-1 is divisible by a-1) ,

      Hence is a composite number.

       
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