Geometric Progression

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.