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 n^th term is given by a_n = ar^n-1 

• The sum, S_n of the first ‘n’ terms of the G.P. is given by S_n = 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, ... ... ..., a_n/b_n will also be in G.P. 

• Suppose a₁, a₂, a₃, ……,a_n are in G.P. then a_n, a_n–1, a_n–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, ……,a_n are in G.P then 1/a₁, 1/a₂, 1/a₃ ……, 1/a_n 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,… ,a_n is a G.P (a_i> 0 ∀i), then log a₁, log a₂, log a₃, ……, log a_n 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, ……,a_n are non-zero positive numbers, then their G.M.(G) is given by G = (a₁a₂a₃ …… a_n)^1/n.

• If G₁, G₂, ……G_n are n geometric means between and a and b then a, G₁, G₂, ……,G_n 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,…, G_n = a(ⁿ⁺¹√b/a)ⁿ.