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Revision Notes on Geometric Progression and Geometric Mean

  • If ‘a’ is the first term and ‘r’ is the common ratio of the geometric progression, then its nth term is given by an = arn-1 

  • The sum, Sn of the first ‘n’ terms of the G.P. is given by Sn = a (rn – 1)/ (r-1), when r ≠1; = na , if r =1

  • If -1 < x < 1, then limxn = 0, as n →∞. Hence, the sum of an infinite G.P. is 1+x+x2+ ….. = 1/(1-x) 

  • If -1 < r< 1, then the sum of the infinite G.P. is a +ar+ ar2+ ….. = a/(1-r)

  • If each term of the G.P is multiplied or divided by a non-zero fixed constant, the resulting sequence is again a G.P.

  • If a1, a2, a3, …. andb1, b2, b3, … are two geometric progressions, then a1b1, a2b2, a3b3, …… is also a geometric progression and a1/b1, a2/b2, ... ... ..., an/bn will also be in G.P. 

  • Suppose a1, a2, a3, ……,an are in G.P. then an, an–1, an–2, ……, a3, a2, a1 will also be in G.P.

  • Taking the inverse of a G.P. also results a G.P. Suppose a1, a2, a3, ……,an are in G.P then 1/a1, 1/a2, 1/a3 ……, 1/an will also be in G.P 

  • If we need to assume three numbers in G.P. then they should be assumed as   a/b, a, ab  (here common ratio is b)

  • Four numbers in G.P. should be assumed as a/b3, a/b, ab, ab3 (here common ratio is b2)

  • Five numbers in G.P. a/b2, a/b, a, ab, ab2  (here common ratio is b) 

  • If a1, a2, a3,… ,an is a G.P (ai> 0 ∀i), then log a1, log a2, log a3, ……, log an is an A.P. In this case, the converse of the statement also holds good.

  • 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 = √ac is the geometric mean of a and c.

  • Likewise, 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 and a and b then a, G1, G2, ……,Gn b will be a G.P.

  • Here b = arn+1, ⇒ r = n+1√b/a, Hence, G1 = a. n+1√b/a, G2 = a(n+1√b/a)2,…, Gn = a(n+1√b/a)n.

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